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EXPLORING A NEW TIME-DEPENDENT METHOD 
FOR MOLECULAR QUANTUM DYNAMICS 



By 

RICARDO LUIZ LONGO 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 
OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 



To 

Dalva, Adhemar, Renata, Milton, and Ivani 



ACKNOWLEDGMENTS 



I wish to acknowledge my advisor Professor Yngve Ohm not only for his counsel, 
but mainly for what he stands for. Also, I would like to thank Dr. Erik Deumens 
and their group, namely, Augie Diz, Juan Oreiro, and Hugh Taylor for their support, 
encouragement, and fruitful discussions. 

I thank Michael Zemer and his group for their friendship in the beginning of my 
time in Lei 212 and the QTP personnel. 

Finally, I could not have so much fun without some of my good friends, like Marshall 
and Genny, Rajiv and Pria, Sylvio, Osiel, Caco, Luciana, Andres, Victor, Monique, Bill, 
Dave, Mark, Charlie. 



lU 



TABLE OF CONTENTS 



ACKNOWLEDGMENTS . . 

LIST OF TABLES 

LIST OF FIGURES 

ABSTRACT 

CHAPTERS 

1 INTRODUCTION . 



2 TIME-DEPENDENT METHODS 5 

PES Based Time-Dependent Methods 6 

PES Free Time-Dependent Methods 9 

Close-coupling Methods 10 

Molecular Dynamics Methods 10 

Time- Dependent Hartree-Fock Methods 11 

3 THE THEORETICAL APPROACH 14 

The END Formalism 15 

Coherent States 15 

The Time-Dependent Variational Principle 20 

The END-SD-FGWP Method 22 

The Equations of Motion 22 

The NDDO Approximation 31 

The END-SD(NDDO)-FGWP Method 34 

AMI Model 37 

Implementation 40 



4 



H+ + H, He, AND H 2 CHARGE TRANSFER COLLISION . . . 

The H-^ + H Collision 

Total Cross Sections 

Differential Cross Sections 

Trajectory Analysis 

Transfer Probability 

Reduced Differential Cross Section 

The H^ + He Collision 

Trajectory Analysis 

The Deflection Function and the Interaction Potential 

Classical Differential Cross Section 

Semiclassical Elastic Differential Cross Section 

Experimental and Theoretical Elastic Differential Cross Sections 

The H^ + H 2 Collision 

Trajectory Analysis 

Vibrational Analysis 

Transfer Probability 

Deflection Function 

Differential Cross Sections 

Summary and Conclusions 



. 44 

. 46 
. 46 
. 50 
. 52 

. 54 

. 59 
. 63 
. 64 

. 67 
. 73 
. 78 
. 85 

. 87 
. 88 
. 92 
. 95 

. 96 
. 97 
. 99 



5 INTRAMOLECULAR CHARGE TRANSFER DYNAMICS 101 

Electron Transfer Formalisms 101 

Structure, Energetics, and Electron Density of LiCNLi 104 

Structure, Energetics, and Electronic Population of LiHLi 110 

Dynamics of Electron Transfer in LiHLi 112 

Summary and Conclusion 117 



6 CONCLUSIONS 119 

REFERENCES 122 



BIOGRAPHICAL SKETCH . 



129 



LIST OF TABLES 



Table 4-1: Contraction coefficients c and exponents a for the basis sets used in this 







Table 4-2: 


Ionization potentials, electron affinities and polarizabilities of H, He and 


Table 4-3: 


Total cross section for + H collision at 500 eV using pVDZ basis set. . 48 


Table 4-4: 


Total transfer cross sections for colliding with a H (x 10"^^ cm^). . 49 


Table 4-5: 


Experimental and theoretical rainbow angles for H'*' + He system. ... 86 


Table 4-6: 


Vibrational frequencies (cm“^) as function of the impact parameter 
(a.u.) and molecular orientation (a°, 0°) 95 


Table 5-7: 


Minima for the linear structure of LiCNLi at various levels of theory. 
Bond distance in pm 105 


Table 5-8: 


Relative energies of minima I and II and of transition state I-II (C*). 
Energies in kcal/mol 106 


Table 5-9: 


Mulliken population (a.u.) and dipole moments (D) of structures I and 
II, transition state I-II and global minimum at UHF/3-21G level. . . 107 



Table 5-10: Mulliken population, structure, and relative energy of the LiHLi 

molecule at SCFAJHF level 113 



VI 



LIST OF FIGURES 



Figure 3-1: 



Figure 4-2: 



Figure 4-3: 



Figure 4-4: 



Figure 4-5: 



Figure 4-6: 



Figure 4-7: 



Figure 4-8: 



Functional diagram of the ENDyne program 42 

Weighted transition probabilities for total electron transfer at 500 eV as a 
function of impact parameter from ENDyne. All data in atomic units. . 49 

Scattering angle as a function of impact parameter using ENDyne, bare 
nuclei, and hydrogen atom potential for the + H system. Energy: 

500 eV. Basis set: pVDZ. Scattering angle in degrees and impact 
parameter in a.u.. Full line: ENDyne, dotted line: bare nuclei, and 
dashed line: hydrogen atom potential 53 

Scattering angle as a function of impact parameter using ENDyne, bare 
nuclei, and hydrogen atom potential for H'^ + H system. Small impact 
parameter region. Energy: 500 eV. Basis set: pVDZ. Scattering angle 
in degrees and impact parameter in a.u.. Full line: ENDyne, dotted line: 
bare nuclei, and dashed line: hydrogen atom potential 54 

Transfer probability versus the scattering angle (in degrees). 

Comparison between the experimental and ENDyne results for the + 

H system. Energy: 250 eV. Basis set: pVDZ. Experimental angular 
resolution ±0.6°, ’+’ ENDyne, from Ref. , and ’x’ from Ref. ... 56 

Transfer probability versus the scattering angle (in degrees). Comparison 
between the experimental and ENDyne results for the + H system. 
Energy: 410 eV. Basis set: pVDZ. Experimental angular resolution 



±0-2° — ±0.6°, ’+’ ENDyne, from Ref. , and ’x’ from Ref 57 

Transfer probability versus the scattering angle (in degrees) for the 
+ H system. Energy: 500 eV. Basis set: pVDZ. ’+’ ENDyne 57 

Transfer probability versus the scattering angle (in degrees). 



Comparison between the experimental and ENDyne results for the + 
H system. Energy: 700 eV. Basis set: pVDZ. Experimental angular 
resolution ±0.02° — ±0.6°, ’+’ ENDyne and from Ref 

• • 

Vll 



58 



Figure 4-9: Transfer probability versus the scattering angle (in degrees). 

Comparison between the experimental and ENDyne results for the + 

H system. Energy: 1(KX) eV. Basis set: pVDZ. Experimental angular 
resolution ±0.07° — ±0.6°, ’-i-’ ENDyne and from Ref. 58 

Figure 4-10: Reduced differential cross sections versus the scattering angle. 

Experimental and the ENDyne results for the -f- H system. Energy: 

250 eV. Basis set: pVDZ, ENDyne transfer, ’+’ ENDyne elastic, 

’ X ’ transfer and ’o’ elastic from Ref. 60 

Figure 4-11: Reduced differential cross sections versus the scattering angle. 

Experimental and the ENDyne results for the H'*' -i- H system. Energy: 

410 eV. Basis set: pVDZ, ’*’ ENDyne transfer, ’-i-’ ENDyne elastic, 

’x’ transfer and ’o’ elastic from Ref. 61 

Figure 4-12: Reduced differential cross sections versus the scattering angle. 

Experimental and the ENDyne results for the -(- H system. Energy: 

5(X) eV. Basis set: pVDZ, ’*’ ENDyne transfer, ENDyne elastic, 

’ X ’ transfer and ’o’ elastic from Ref. 61 

Figure 4-13: Reduced differential cross sections versus the scattering angle. 

Experimental and the ENDyne results for the -(- H system. Energy: 

700 eV. Basis set: pVDZ, ’*’ ENDyne transfer, ’-i-’ ENDyne elastic, 

’ X ’ transfer and ’o’ elastic from Ref. 62 

Figure 4-14: Reduced differential cross sections versus the scattering angle. 

Experimental and the ENDyne results for the + H system. Energy: 
1(X)0 eV. Basis set: pVDZ. ’*’ ENDyne transfer, ’+’ ENDyne elastic, 

’ X ’ transfer and ’o’ elastic from Ref. 62 

Figure 4-15: Dynamical trajectories of being scattered by He. ENDyne results at 

energy of 50.0 eV with a pVDZ basis set 65 

Figure 4-16: Dynamical trajectories for scattering angle of 2.65°. ENDyne results for 

H"'’ + He system at 50.0 eV with a pVDZ basis set. Impact parameters: 
solid line ( — ) = 1.12 a.u., dashed line ( — ) = 1.60 a.u., dotted line (• • •) 

= 2.10 a.u 66 



VUJ 



Figure 4-17: Motion of the atomic target (He) during a collision with H"*" at 50.0 eV 

as computed by ENDyne with a pVDZ basis set. The initial coordinate 
of the He atom is (15.0, 0.0). Impact parameters: solid line ( — ^) = 1.12 
a.u., dashed line ( — ) = 1.60 a.u., dotted line (• • •) = 2.10 a.u 67 

Figure 4-18: Deflection function for H"^ + He. ENDyne results at 50.0 eV with a 

pVDZ basis set 68 

Figure 4-19: Potential energy curve obtained from the inversion of the deflection 

function generated by ENDyne for H"^ + He at 50.0 eV with a pVDZ 
basis set 70 

Figure 4-20: Potential energy curve obtained from the direct time evolution. ENDyne 

results for H"^ + He at 50.0 eV with a pVDZ basis set 71 

Figure 4-21: Potential energy curves for H"^ + He obtained from a CIS calculation 

with a pVDZ basis set 72 

Figure 4-22: Deflection function for H'^ + He. ENDyne results at 500 eV with a 

pVDZ basis set. Regions: I) diamonds ’o’, II) triangles ’a’, and. III) 
open circles ’o’ 74 

Figure 4-23: Elastic differential cross section for H"^ + He. ENDyne results at 500 

eV with a pVDZ basis set. Regions: I) diamonds ’o’, II) triangles ’a’, 
and, in) open circles ’o’ 75 

Figure 4-24: Charge transfer differential cross section for H^ + He. ENDyne results 

at 500 eV with a pVDZ basis set. Regions: I) diamonds ’o’, II) 
triangles ’a’, and, IE) open circles ’o’ 76 

Figure 4-25: Elastic and charge transfer differential cross sections for H'*' + He. 

ENDyne results at 500 eV with a pVDZ basis set 77 

Figure 4-26: Elastic and charge transfer differential cross sections for H"*^ + He. 

ENDyne results at 1500 eV with a pVDZ basis set 77 

Figure 4-27 : Elastic and charge transfer differential cross sections for H"^ + He. 

ENDyne results at 5000 eV with a pVDZ basis set 78 



IX 



Figure 4-28; Elastic differential cross section for H+ + He. Energy: 5000 eV. Basis 

set: pVDZ. Solid line corresponds to the semiclassical corrections to the 
ENDyne results. Open circles are experimental data 87 



Figure 4-29: Dynamical trajectories for H'*’ + H2 collision from ENDyne calculation 



for (0°, 0°) orientation at 500 eV with a pVDZ basis set 89 

Figure 4-30: Dynamical trajectories of the target Hj. ENDyne results for (0°, 0°) 

orientation at 500 eV with a pVDZ basis set 90 

Figure 4-3 1 : Dynamical trajectories for + H2 collision from ENDyne calculation 

for (90°, 0°) orientation at 500 eV with a pVDZ basis set 91 

Figure 4-32: Dynamical trajectories of the target H2. ENDyne results for (90°, 0°) 

orientation at 500 eV with a pVDZ basis set 91 

Figure 4-33: Interatomic distances of the target H2 as function of time. ENDyne 

results for (0°, 0°) and (90°, 0°) orientations at 500 eV with a pVDZ 
basis set. (1 a.u. = 0.024195 fs) 93 

Figure 4-34: Mbrational spectrum of the target H2. ENDyne results for (0°, 0°) 

orientation with impact parameter of 1.25 a.u. Energy: 500 eV. Basis 
set: pVDZ 94 



Figure 4-35: Probability for charge transfer in -1- H2 collision. ENDyne results for 

(0°, 0°) and (90°, 0°) orientations. Energy: 5(X) eV. Basis set: pVDZ. . 96 

Figure 4-36: Deflection functions for + H2 collision. ENDyne results for (0°, 0°) 

and (90°, 0°) orientations. Energy: 5(X) eV. Basis set: pVDZ 97 

Figure 4-37 : Elastic differential cross section for + H2 collision. ENDyne results 

for (0°, 0°) and (90°, 0°) orientations. Experimental data from reference 
. Energy: 5(X) eV. Basis set: pVDZ 98 

Figure 4-38: Charge transfer differential cross section for H"*" -1- H2 collision. 

ENDyne results for (0°, 0°) and (90°, 0°) orientations. Experimental 
data from reference . Energy: 500 eV. Basis set: pVDZ 98 



Figure 5-39: Diabatic and adiabatic potential curves for normal electron transfer. . 102 



Figure 5-40: Transition state for structures I and II and global minimum structures. 

Bond distances in pm and angles in degrees 105 

Figure 5-41: Alpha and beta isodensity for structure I at UHF/3-21G level. 

Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). Structure: Li-C = 197.98 
pm (= 1.980 A = 3.741 a.u.), C-N = 115.02 pm (= 1.150 A = 2.174 
a.u.), Li-N = 192.59 pm (= 1.926 A = 3.639 a.u.). SCF energy: 
-106.63313290 a.u. = -2901.6368 eV 108 

Figure 5-42: Alpha and beta isodensity for structure II at UHF/3-21G level. 

Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). Structure: Li-C = 214.10 
pm (= 2.141 A = 4.046 a.u.), C-N = 115.77 pm (= 1.158 A = 2.188 
a.u.), Li-N = 178.79 pm (= 1.788 A = 3.379 a.u.). SCF energy: 
-106.63909258 a.u. = -2901.7989 eV 108 

Figure 5-43: Alpha and beta isodensity of the transition state I-II structure (Cs) at 

UHF/3-21G level. Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). 
Structure: Li-C = 208.75 pm (= 2.088 A = 3.945 a.u.), C-N = 115.83 
pm (= 1.158 A = 2.189 a.u.), Li-N = 181.89 pm (= 1.819 A = 3.437 
a.u.), Li-C-N = 148.49°, C-N-Li = 138.61°. SCF energy: 

-106.63244750 a.u. = -2901.6180 eV 109 



Figure 5-44: Alpha and beta isodensity of the global minimum structure (C 2 v) at 

UHF/3-21G level. Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). 
Structure: C-N = 117.46 pm (= 1.175 A = 2.220 a.u.), Li-N = 189.22 



pm (= 1.892 A = 3.576 a.u.), C-N-Li = 137.12°. SCF energy: 
-106.64697039 a.u. = -2902.0134 eV 109 

Figure 5-45: Mulliken charge differences at SCFAJHF level in the Li(l)-H-Li(2) 

molecule as function of relative bond distances (ri - r 2 ) Ill 



Figure 5-46: Potential energy curves of the Li(l)-H-Li(2) molecule at SCFAJHF 

level 



112 



Figure 5-47: Bond distances, ri = distance Li(l)-H and T 2 = distance H-Li(2) in a.u. 



as a function of time. ENDyne calculation with a H/3— 21G and 
Li/3-2 1-Kj basis set 114 

Figure 5-48: Atomic Mulliken population of Li(l) and Li(2) as a function of time. 

ENDyne calculation with a H/3-21G and Li/3-2 1-K3 basis set 115 



Figure 5-49: Atomic Mulliken charge differences q[Li(l)]-q[Li(2)] as a function of 

time. ENDyne calculation with a H/3-21G and Li/3-21+G basis set. 116 

Figure 5-50: Alpha and beta spin atomic Mulliken populations of Li(l) and Li(2) in 

the 15% structure as a function of time. ENDyne calculation with a 
H/3-21G and Li/3-2 1-K5 basis set 117 



XU 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 

EXPLORING A NEW TIME-DEPENDENT METHOD 
FOR MOLECULAR QUANTUM DYNAMICS 

By 

RICARDO LUIZ LONGO 
May, 1993 

Chairman: N. Yngve Ohm 
Major Department: Chemistry 

The electron-nuclear dynamics in chemical processes is described by a method 
founded on the Time-Dependent Variational Principle (TDVP). In addition, to avoid 
redundancies in the parametrization of the electronic wave function, coherent states (CS) 
are used. The equations resulting from this treatment enable us to describe the dynamics 
in general molecular system, since it does not presume a priori knowledge of the potential 

energy surface of the molecular system. The theory is called Electron Nuclear Dynamics 
(END). 

In order to make this time-dependent method based upon TDVP and CS practical 
and still realistic, we treat the nuclei in the limit of narrow Gaussian wavepackets (which 
corresponds to classical nuclei) and use a single determinant to describe the electrons. 
We name this approach END-SD-FGWP. 

We use the END-SD-FGWP approach to study the dynamics of charge transfer in 

• • • 

XUl 



the systems Li^-H— Li ^ Li— H-Li^ and Li+-CN-Li ^ Li-CN-Li-". The collision of a 
proton with H, He, and H2 is also investigated by the END-SD-FGWP approach. Several 
properties of these collisions are compared directly with the experimental results. 

Since we are interested in the application of the END formalism to large systems we 
propose one more approximation, namely, the neglect of the differential diatomic over- 
lap (NDDO), and introduce the END-SD(NDDO)-FGWP approach. The MNDO/AMl 
parametrization is used for the NDDO approach. The computational savings are twofold: 
a) the number of integrals computed are reduced by several orders of magnitude com- 
pared to the ab initio approach, b) the equations of motion are simplified, and c) the high 
frequency motions of the core electrons is removed by an effective core. 



XIV 



CHAPTER 1 
INTRODUCTION 



Time-dependent methods for solving the Schrodinger equation were introduced in 
quantum mechanics since its beginning. For instance, already in the 1930s Dirac sug- 
gested the time-dependent Hartree-Fock method [1]. However, due to the theoretical 
and computational complexity of time-dependent methods, their development and ap- 
plication to molecular systems have been delayed by several decades in comparison to 
time-independent methods. Despite these difficulties, time-dependent methods for molec- 
ular quantum dynamics have gained in importance in the last two decades. Certainly, 
the interest in time-dependent solutions of the Schrddinger equation is very much due to 
the development in recent years of novel experimental techniques that make it possible 
to measure quantum state-specific properties of rather complicated systems and to probe 
the dynamics of chemical reactions at the molecular level for very short times. However, 
a solid theoretical understanding of non-stationary processes is highly desirable, and due 
to the development of new computational techniques and powerful computer hardware, 
assumptions, hypotheses and theories describing these processes can now be explored. 

The time-dependent approach to quantum dynamics offers a number of advantages 
over time-independent methods. For example, in a reactive scattering process, plots of 
the particle density as it evolves throughout the collision are easily interpreted using 
simple concepts of the underlying physics. These plots can be used to improve one’s 
understanding of mechanisms that are important during a reaction or in other physical- 



1 



2 



chemical processes. From a theoretical point of view, the time-dependent approach makes 
it straightforward to include the effect of intense laser fields or external time-dependent 
heat baths, where the dynamics is treated approximately by classical mechanics, statistical 
or stochastic methods. It is also well adapted to improvement in the dynamics for those 
degrees of freedom that are described classically, by the wavepacket propagation, as 
well as those given by explicit quantum mechanical treatments. As a result, the END 
and other time-dependent approaches offer great flexibility for the choice of various 
approximations and is also capable of a dynamically accurate description. It is also 
possible to use within these approaches some of the experience acquired for decades 
with time-independent methods to provide a practical and realistic description of the 
chemical system while at the same time keeping an accurate description of the dynamics. 
It also allows the blending of classical, wavepacket, quantum operator, state expansion 
and numerical techniques necessary for solving complex dynamical problems. 

We present a time-dependent method for solving the Schrddinger equation and explore 
its capabilities in describing charge exchange collisions, molecular vibrations, and electron 
transfer dynamics. The method presented here attempts to describe the simultaneous 
electron and nuclear dynamics. It has been developed during the last half decade and has 
the very nice feature of not requiring potential energy surfaces of the molecular system. 
It allows both electrons and nuclei to be treated quantum mechanically. The dynamical 
equations of motion are obtained using the time-dependent variational principle (TDVP) 
and the wavefunctions are parametrized using coherent states (CS). The combination of 
TDVP/CS gives rise to a new time-dependent approach to molecular dynamics called 



3 



Electron Nuclear Dynamics (END). The END formalism provides dynamical equations 
that look simple and are easy to interpret. The computer implementation of this approach 
is not trivial, and in order to make it practical some approximations are necessary. Based 
upon the experience acquired with time-independent methods, the nuclei are treated 
classically or, more appropriately, in the limit of narrow Gaussian wavepackets, and 
the electrons are described by a single determinant. The theory for quantum nuclei and 
multi-determinantal description of the electrons has already been developed [2], but its 
implementation is not yet done. So we think it would be interesting and fruitful to explore 
in some detail the single determinant/classical nuclei model. Certainly, this simplest 
model for the END formalism may be inadequate for describing some problems, mainly 
the ones involving many product channels (multi-reference character) and tunneling 
dominant processes [3]. However, since the END formalism does not require the 
knowledge of the potential energy surface(s) of the molecular system it can be applied to 
a large variety of systems and dynamical problems in an unified manner. For instance, 
it can be used to study scattering problems (total, differential and state to state reactive 
cross sections), charge transfer dynamics (rates and mechanisms), molecular vibrations 
(interpretation of transition state spectroscopy experiments and prediction of reaction 
rates). It permits us to follow chemical reactions, predict and interpret Raman spectra, and 
many other dynamical processes. Future implementation of time-dependent field in the 
molecular Hamiltonian would make the END formalism able to describe photodissociation 
processes, laser chemistry, dynamical nuclear magnetic resonance (NMR), etc. It should 
be noted however, that a fully ab initio treatment of large molecular systems using 



4 



the END/single determinant/classical nuclei approach is not yet practical. Therefore, in 
addition to exploring the ab initio version for describing reactive collisions, molecular 
vibrations, and electron transfer in small systems, we propose to employ the neglect of 
the differential diatomic overlap (NDDO) approximation [4]. It is our expectation that 
the NDDO approximation on top of the END/single determinant/classical nuclei model 
will decrease the computational demand in many ways. 

a) The number of molecular integrals and their gradients involved would be several orders 
of magnitude smaller than the ab initio version. 

b) Since it is commonplace to use effective potential for the core electrons it should, in 
addition to decreasing the number of degree of freedom (differential equations needed 
to be solved) also eliminate the high frequency motions of the core electrons. This will 
allow the integrator of the differential equation to take larger steps decreasing by several 
orders of magnitude the time to solve the differential equations. 

c) In general, minimum basis sets are the working frame for the NDDO approximation, 
which decreases significantly the number of equations of motion. 

The outline of this work is as follows. Chapter 2 describes some of the available 
time-dependent methods for molecular dynamics. The END formalism is introduced in 
Chapter 3 with the NDDO approximation and concluded with the END-NDDO equations. 
In Chapter 4 we present the results, analysis and discussion of the reactive scattering of 
on H, He, and H 2 . Chapter 5 is devoted to the study of the intramolecular charge transfer 

systems y-CN-Li ^ Li-CN-Li-" and y-H— Li ^ Li— H-Li-". The conclusions are 
presented in Chapter 6. 



CHAPTER 2 

TIME-DEPENDENT METHODS 



The existing time-dependent methods that are relevant to our work are described in 
this chapter. The aim is to provide a general view of these methods with emphasis on 
performance, limitations, and comparisons among them. It seems that the best way to give 
an overview of these time-dependent methods is to provide some kind of classification 
that compare their main assumptions. In order to provide such a classification, the concept 
of potential energy (hyper)surface (PES) is fundamental. For practical purposes a PES 
is a function that gives the value of the electronic energy of a group of N interacting 
atoms with respect to their 37V — 6 relative position coordinates. For each electronic 
state of the molecular system we have one PES which contains information about 
the isolated reactant and product species, their long-range interactions, their molecular 
deformations which lead to the breaking and forming of chemical bonds, the geometries 
and properties of transient reaction intermediates, and energy barriers for going from one 
of those intermediates to another. As we can see, the task involved in obtaining such 
a (hyper)surface for polyatomic molecules is tremendous. Highly correlated electronic 
structure calculations with very large basis sets are necessary in order to provide an 
accurate description of the different regions of the PES. Once the PES has been mapped, 
a fitting procedure will be employed to give the functional form of the PES. This fitting 
procedure is very time consuming and tedious, and it is still more an art than a scientific 
procedure. The functional form of a PES can be quite complicated. As evidence for that 



5 



6 



we can mention that even for a triatomic molecule like H3, an attempt to use the well 
developed algebra program (REDUCE) to evaluate the first, second, third, and fourth 
derivatives analytically failed due to the algebraic complexity [5]. We classify the time- 
dependent methods into two categories a) PES based methods, and b) PES free methods. 

PES Based Time-Dependent Methods 

Once the potential energy surface for the molecular system has been determined, 
there are basically three approaches to perform dynamics on this surface, namely, a) 
the classical trajectory method, b) the semiclassical method, and c) the quantum time- 
dependent method. The given order a), b), and c) is related to the increasing theoretical 
and implementation difficulties, computational demand, and accuracy. 

The widely used classical trajectory approach [6-8] employs Newton’s equation 
of motion to treat the dynamics of the nuclei on the PES. This approach is easily 
implemented and the interpretation of the results are simple. However, the computational 
demand can be large for polyatomic systems, and a good statistical analysis program is 
necessary. A detailed classical trajectory simulation requires an extensive sampling in 
the phase space (50 000 to 100 000 trajectories for a triatomic system) for each initial 
condition (collision energy, vibrational and rotational initial states). This computational 
demand is not the only drawback of the classical method. It neglects quantum effects like 
tunneling, interference, penetration (generalized tunneling), zero-point motion, resonance 
behavior, etc, which can be fundamental for chemical encounters. These deficiencies 
represent an inherent obstacle for the classical trajectory approach to provide qualitative 
or quantitative results for chemical processes. Despite these problems, the classical 



7 



trajectory approach has been applied to a widely variety of dynamical problems including 
scattering of atoms or molecules by solid surfaces [9] and bimolecular reactions [8], In 
most of these applications (semi)empirical potential energy surfaces are used, such as 
DIM (Diatomic-in-Molecules), MIM (Molecule-in-Molecules), LEPS (London-Eyring- 
Polany-Sato), etc [10, 11]. 

Semiclassical methods have been designed to overcome some of the flaws in the 
classical description. They assume that the quantum effects can be treated as corrections 
to the classical motion, based on the fact that the nuclei are relatively heavy compared 
to the electronic mass. The semiclassical approaches have, in principle, the advantage 
of incorporating some of the important quantum effects while maintaining the simple 
implementation. However, the generalization of the semiclassical methods is not simple, 
and, as result, each system requires a special implementation, and the validity of the 
approximations is often not clear. The perhaps most developed semiclassical dynamical 
method has been proposed by Heller [12—15] and employs the Gaussian wavepacket 
approximation. That is, the quantum wavepacket is constrained to be of a Gaussian 
form. This semiclassical method has been widely used in many applications with some 
success [12-15]. Another well-developed semiclassical method that uses the so called 
eikonal approach to dynamics has been proposed by Micha [16]. Comparisons of the 
results provided by these two semiclassical methods, namely the Gaussian wavepacket 
and the eikonal approach, have shown that they differ only in some minor details [17, 18]. 
These semiclassical methods have been applied to describe photodissociation processes, 
reactive scattering and bimolecular reactions [15]. 



8 



Among the various quantum time-dependent methods on PES two have been widely 
used, namely, the ’exact’ time-dependent quantum mechanical and the quantum time- 
dependent self-consistent field (TDSCF) methods [19, 20]. Even the so called ’exact’ 
time-dependent methods do use some approximations, which lead to errors that can, in 
principle, be made as small as desired. The ’exact’ time-dependent methods can be 
classified according to the representation of the wavefunction and the algorithm for 
the time evolution. We distinguish two approaches to wavefunction representation, 
namely, discretization on a grid of points, sometimes referred to as DVR (discrete 
variable representation), and expansion in a basis set. In general, the numerical cost 
of the discretization and basis set expansion representations scales as 0{V^) and 0{N^), 
respectively, where V is the volume of phase space containing the system and N is the 
number of expansion functions. There are basically four algorithms in use for the time 
evolution: SOD (second-order differencing), SO (split operator), SIL (short-time iterative 
Lanczos), and CE (Chebychev expansion). For a detailed presentation and comparisons of 
the different ’exact’ time-dependent methods, see references [20, 21]. Applications of the 
’exact’ time-dependent methods have been restricted to at most three degrees of freedom 
and include mainly photodissociation processes, absorption and photodetachment spectra, 
reactive scattering (atom-diatom), and unimolecular reactions [22]. The TDSCF method 
assumes variables separation such that each degree of freedom is treated in the mean field 
of the remaining degrees of freedom. In the static SCF this approximation, also called 
the independent particle model, leads to the lack of correlation between the degrees of 
freedom of the system. However, in the case of TDSCF method energy is allowed to flow 



9 



between degrees of freedom leading to some correlation among them. As a result, the 
TDSCF scheme is expected to be accurate for calculations of such averaged properties as 
lifetimes, mean fragment energies, energy flows, etc. In fact, the dissociation dynamics 
of the l 2 He van der Waals complex has been examined by ’exact’ and TDSCF methods. 
It has been shown that for this system the TDSCF scheme gives results in excellent 
agreement with ’exact’ method when the calculated properties are obtained during a 
period of time much larger than the vibrational periods and so long as the excitation 
level is not too high. The TDSCF approach also reproduces well the dephasing events 
in the dynamics and gives very good results for the autocorrelation function [19, 20]. 

We can summarize this section by saying that the greatest drawback of these methods 
(classical, semiclassical, ’exact’, TDSCF) is the need for one or more potential energy 
surfaces. In the case of classical methods the lack of any quantum effects can cause 
problems, the semiclassical approaches are too system specific, and the ’exact’ schemes 
are very numerically intensive and are restricted to problems with few degrees of freedom. 
No doubt these methods have their advantages and their contribution to the understanding 
of time-dependent processes must be emphasized. 

PES Free Time-Dependent Methods 



We focus in this section on time-dependent methods that do not require potential 
energy surfaces. Among these methods we intend to discuss the time-dependent methods 
that fall into these three classes: a) close-coupling, b) molecular dynamics, and c) time- 
dependent Hartree-Fock. 



10 



Close-coupling Methods 

The close-coupling methods [23-25] consist in projecting the time-dependent 
Schrodinger equation on a basis. The electronic states are described by a Cl (config- 
uration interaction) expansion where the time-dependence is put into the Cl coefficients. 
The nuclear dynamics are not treated explicitly and are constrained to follow a priori 
trajectories, such as linear or Coulombic trajectories. 

The description of the electronic states by a Cl wavefunction imposes a heavy 
computational demand restricting the application of the close-coupling methods to simple 
one electron (or pseudo one-electron) systems. Recent extensions to two-electron systems 
seems not to be very practical [26]. 

The prescribed trajectories for the nuclei limit the application of the close-coupling 
methods to high energy (above 1.0 keV) collisions. It also makes the application of 
these methods for the calculation of differential cross sections other than the smallest 
scattering angles quite doubtful. 

Molecular Dynamics Methods 

Molecular dynamics (MD) methods assume that the atomic dynamics is governed 
by the laws of classical mechanics. That is, MD methods neglect quantum effects on 
the nuclear motion and the electrons are considered as being always in the ground- 
state corresponding to the instantaneous nuclear configuration. As a result the explicit 
knowledge of the interatomic potential is required. Obtaining this interatomic potential is 



11 



a difficult many-body problem and different ways of calculating this potential distinguish 
the different MD methods. 

The most widely used MD methods are the ones that employ (semi)empirical force 
fields for describing the interatomic potential [27]. These MD/force-field methods can 
be combined with perturbation methods to provide thermodynamical properties [28], in 
addition to structural (geometric and conformational) information. They can be applied 
to a great variety of systems including large biological molecules [29]. 

Some of the other MD formulations evolve computing the interatomic potential 
by solving the Schrddinger equation in some approximate manner (e.g. ab initio or 
semiempirical Hartree-Fock (HF), density functional (DF) theory, etc) for each nuclear 
configuration during the time evolution [30, 31]. 

Time-Dependent Hartree-Fock Methods 

Despite the fact that the time-dependent Hartree-Fock (TDHF) method was suggested 
by Dirac already in 1930s [1] only recently has this method been developed and applied 
to reactive and inelastic scattering [32-35] and molecular problems [36, 37]. 

In TDHF the quantum dynamics of the electrons represented by a single Slater 
determinant is coupled to a classical description of the nuclear dynamics in a self- 
consistent fashion. The differences between some of the TDHF implementations are 
related to the way the Hartree-Fock equation is approximately solved and to the way that 
the translation of the basis set with the nuclei is handled. This is related to the electron 
translation factor (ETF) problem. 



12 



The NDDO approximation has been used to simplify the TDHF equations and applied 
in the calculation of molecular properties, such as atomic charges, structure, dipole 
moments, etc, during time evolution of LiH, H 2 O, and CH 2 O molecules [36]. The density 
functional theory (DFT) using the local density approximation (LDA) has been employed 
to compute the Hartree-Fock potential for the TDHF method. This approach has been 
used in the dynamics of Na 4 cluster for the singlet and triplet electronic states [37]. Both 
approaches, namely, TDHF-NDDO and TDHF-DFT, do not consider the effects of the 
translation of the basis set with the nuclear motion upon the electron dynamics. This 
means that the electron translation factors are not properly taken into account by these 
approaches, neither are the electron-nuclear couplings. 

The proper derivation of TDHF equations with the inclusion of the electron translation 
factors has been done using the density matrix instead of the orbital coefficients as the 
time -dependent quantities [34]. However, no applications of this method have yet been 
made, and only after the neglect of the electron translation factors (RTF’s) was this 
approach applied to the H'*’ + H collision [34]. It is expected that in a short time 
applications including the ETF’s will appear and we shall be able to see the magnitude 
of the errors caused by the neglect of these effects. 

In summary: 

1 . PES based time-dependent methods have the drawback of requiring a priori knowl- 
edge of the interparticle potentials. This requirement limits the applicability of these 
methods to small systems (just a few degrees of freedom) and low energy processes 
(high energy implies in a large number of excited PES’s), 



13 



2. PES free time-dependent methods are very promising, but general computer codes, 
extensive testing of the approximations, and a solid foundation of the formal devel- 
opments are still required. 

As a conclusion to this chapter we can say that there are still a lot of formal 
and numerical investigations to be performed on time-dependent methods for molecular 
quantum dynamics. In the next few chapters we shall be able to contribute to the 
investigation of a new time-dependent method. 



CHAPTER 3 

THE THEORETICAL APPROACH 



In the last chapter we expressed our concern for potential energy surface (PES) based 
time-dependent methods being limited to systems where these PES’s are known. In this 
chapter we present the equations for a recently proposed time-dependent approach to 
molecular dynamics called Electron Nuclear Dynamics (END). 

The time-dependent variational principal (TDVP) is used to obtain an approximate 
time evolution of the molecular system. In order to obtain the equations of motion it is 
necessary to decide how the electrons and nuclei will be treated. From the experience 
with time-independent methods and some of the time-dependent methods, like TDSCF 
and TDHF, it seems that treating the electrons at the single determinant level with classical 
nuclei should provide a very realistic approach for a wide variety of chemical processes. 
Since the classical nuclei approximation can be obtained as the narrow width limit of 
a frozen Gaussian wave packet treatment, we denote the method resulting from these 
approximations as END-SD-FGWP. That is, END stands for Electron Nuclear Dynamics 
from the approximate dynamical equations obtained from the TDVP, SD is the Single 
Determinantal description of the electrons, and FGWP is the classical, or narrow width 
limit of Frozen Gaussian Wave Packet representation of the nuclei. The coherent state 
(CS) for the electrons is the Thouless single determinant representation, which gives a 
suitable parameterization. 

In order to make the END-SD-FGWP approach suitable for studying dynamics of 



14 



15 



large molecular systems, we employ the neglect of diatomic differential overlap (NDDO) 
approximation. 

We have divided this chapter into five sections, namely, the END formalism, the 
END-SD-FGWP method, the NDDO approximation, the END-SD(NDDO)-FGWP model, 
and implementation. In the first section the CS, the TDVP, and the dynamical equations 
are introduced. In the NDDO section we have a subsection where we present the Austin 
model 1 (AMI) realization of the NDDO approximation. 

The END Formalism 

We present a formalism that treats the electron nuclear dynamics without the need of 
a priori knowledge of potential energy surfaces or interatomic potentials. In other words, 
in this formalism only the Hamiltonian (e.g., Coulombic) and the initial conditions (initial 
nuclear positions and momenta, for instance) are needed for the calculation to proceed. 
This formalism relies upon the time-dependent variational principal (TDVP) of a quantum 
state describing the system through a group theoretical coherent state (CS). 

In the next two subsections we present the dynamical equations for a molecular 
system derived using the TDVP and the CS representation. The actual derivation has 
been presented elsewhere [38, 39] and will not be repeated here. 

Coherent States 

We now present the parametrization of the wavefunction in terms of coherent states 
(CS). A coherent state [38, 40] of a molecular wave function |C) is a continuous function 
of the labeled set of electronic and nuclear parameters (. This set of parameters has a 



16 



positive measure d( such that the resolution of the identity 

j ICHCWC = 1 



(3.1) 



holds. 

This choice of representing the wave function is very important since it removes any 
parameter redundancies and leads to a simple interpretation of the dynamical equations. 
The removal of the redundant parameters is essential to avoid complications with the 
phase space that could lead to artificial singularities and difficulties with integration. A 
coherent state representation also leads to a natural division of the parameters into (gen- 
eralized) canonical coordinates and conjugate momenta. Thus, they form a generalized 
phase space with the expectation value of the energy being the Hamiltonian function for 
the time evolution of the parameters. As a result, the equations for the CS parameters 
form a classical Hamiltonian system facilitating their physical interpretation. 

The generalization to a multi-determinantal description would follow the same general 
approach and has been done elsewhere [2] . We are interested in those coherent states that 
are related to compact Lie groups and to a single determinant. Using the group theoretical 
approach we can eliminate the redundancy of the single determinant representation. Let 
us assume that we have an orthonormal spin orbital basis set made up of K vectors 

and N particles which will be described by a single determinant. This 
determinant can be found as a unitary transformation acting on the orbital expansion 
coefficients of some reference single determinant |^o)- We define this reference state 



17 



l^o) = n^^K^) (3.2) 

/=1 

where b] (6,) are the fermion creation (annihilation) operators of the spin orbital basis 
used. The reference state is a lowest weight state for the irreducible representation 
[l'^0(^“^)] of the group U{K) with generators bjbj because the weight or number 
operators b]b{ have eigenvalues 1 for * = 1, . . . , iV and 0 for i = iV + 1, . . . , ii'. As we 
can see we have divided our set of K spin orbitals into two subsets N and K — N. We 
now denote any quantity q as: 

1. q if q refers to the set of all K spin orbitals; 

2. q* if q refers to the subset of N spin orbitals; and 

3. 9° if q refers to the subset of K - N spin orbitals. 



In the context of a determinantal state this division of the set of K spin orbitals into 
the two subsets: N and K — N takes the meaning of occupation, where the subset N 
is called occupied and K — N is the unoccupied or virtual subset. In the general case, 
however, these subsets N and K — N will be bullet and open circle, respectively. Also, 
the following notation will be used when dealing with matrices: 

/ “11 ••• ^IN “l(7V+l) ••• ^'iK \ 



u = 



u 



^N1 

(AT+1)1 






u 



n 

K\ 



u 



u 

n 



NN 



u 



It 

KN 



u 



N{N^\) 



(TV-fl)A^ ^(Ar-hi)(AT+i) 



u 



A'(iV+l) 



u 



NK 



u 






u 



KK 



(3.3) 



^NxN 

If" 

^K-NxN 



^NxK-N \ _ 
^K-NxK-N 




• • • 



/ 



18 



We now assume that there is an t/ matrix which transforms the spin orbital basis 
to the occupied and virtual spin orbitals, such that, all redundancies due to unitary 
transformations among the occupied (virtual) spin orbitals are removed, that is. 






( 3 . 4 ) 



Then, the coherent state representation of a determinantal state takes the form: 



N 






1=1 



= n ( E + E 1 1'"“^) 

i=l \i=l ji=7V+l 



N / N 



K N 



= n I E 1 + E E f'yikVtr' I f'l* I \o“c) 

j=N+l k=l 



i=l \/=l 



( 3 . 5 ) 



“HK+ E 

i=l V >=A^+lib=l 



=a 



n('+ E 

i=i \ y=jv+iJb=i / 1=1 



19 



and since we work with an unnormalized coherent state we have that 




N K 



=n n 



t=l j=N-\-l 



(3.6) 



N K 






1=1 j=N+l 




where ^ 




The unnormalized determinantal wavefunction can be expressed in terms of the 
orthonormal spin orbital basis set and the dynamical variables {zji,Zji} 

in the following way 



where the dynamical spin orbitals 

K 

Xi = V’i + (1 < * < N) (3.8) 

j=N+i 

are nonorthogonal, but linearly independent. The corresponding unoccupied dynamical 
orbitals may be chosen as 



and although mutually nonorthogonal they are orthogonal to the occupied space. 



\z) = det{x,} 



(3.7) 



N 




(3.9) 



20 



The Time-Dependent Variational Principle 



We now give an outline of how the time-dependent variational principle (TDVP) can 
be used to provide us with an approximate time evolution of the quantum mechanical 
system. There are several forms of the TDVP [1, 41, 42], but they are all equivalent 
when the trial wave function is described by complex parameters and is analytic in this 
parameter space [43]. 

The TDVP approach consists in minimizing the variations of the quantum mechanical 
action functional A, that is. 




f[(Cl|(IO) 



dt 



«ci)ic) - (ci^io Mcicr' = 0 



(3.10) 



where \Q is quantum state of the system labeled with a set of electronic and nuclear 
parameters ( and H is the Hamiltonian of the system. The TDVP approach leads then 
to the following equation for the time-dependent state |C) 




(3.11) 



This is the time -dependent Schrodinger equation, provided that \6() is allowed to vary 
over all Hilbert space and the right-hand side vanishes. It can be shown [38] the right- 
hand side of equation (3.11) vanishes when we consider explicitly the overall phase 
factor of the time-dependent state |(). 



21 



The dynamical equations resulting from this approach are [38, 39] 






E = 

0 



dE 

d(o 



(3.12) 



which in matrix form becomes 



C 0 
0 -C* 




c 

c* 



/ dE 

( w 

\ M 

V ac 



(3.13) 



and the elements of the metric matrix defined as 

^ a^InS 

^010 — p,. 



(3.14) 



dQ9C0 

where the energy of the system is given by the expectation value of the Hamiltonian 



H as follows 



E{C\0 = 



{C\H\0 

(CIO 



(3.15) 



and the norm is 



5(C*,C) = (CIC). 



(3.16) 



It also should be mentioned that in this formulation the global phase factor exp (^ 7 ) of 
the wavefunction 



l^') = exp (n) 10 



(3.17) 



has been factorized from the evolving state and can be obtained by a simple quadrature 
integration of the following differential equation 



7 4E(<. 

a 



a,n5(c,o_^.ai^ 






\ CK 



dCa 



d\nSiC,0 

dCa 



da 



(3.18) 



-^(C*,C) 



22 



during the evolution. This global phase should be important in applications where the 
time-correlation functions need to be computed. 

The END-SD-FGWP Method 

We have so far obtained dynamical equations that describe the dynamics of a general 
quantum molecular system. As mentioned before, in order to obtain equations of motion 
for the electron nuclear dynamics that are practical and realistic, some approximations 
are needed. We choose a model for which 

(1) the electrons are described by a single Slater determinant, and 

(2) since we would like to make use of the available computer codes for molecular 
integrals we use a truncated spin orbital basis without electron translation factors (ETF’s). 
In other words, the basis set is independent of the nuclear velocity (or momentum), and 

(3) use a classical description of the nuclei. In mathematical terms, this means that when 
employing the TDVP approach the limit of narrow Gaussian wave packets is taken. As 
a result, the nuclear parameters become the positions and momenta of the nuclei. 

TTiese approximations allow us to obtain the equations of motion for END-SD-FGWP 
method. 

The Equations of Motion 

Taking the limit of narrow Gaussian wave packets or the classical description of the 
nuclei permit us to define the dynamic variable for the nuclei as [38, 39] 



Zk = Rk + iPk 



(3.19) 



23 



where R and P are the nuclear coordinates and momenta, respectively. Since we have 
assumed that the dependence of the basis on the nuclear parameters is only through the 
nuclear positions, the equations of motion for the electron and nuclei become [38, 39] 



/ iC 


0 


iCR 


0 ^ 








(dEldz*\ 


0 

j 


-iC* 




0 




• ♦ 




dE/dz 


iC R 


-iCl 


Crr 


-I 




k 




dEjdR 


0 


0 


I 


0 ) 




\p) 




\ dEjdP J 



where 




d‘^\nS{z\R\P\z,R,P) 

dz*dXk 



R'=R,P'=P 



and 



CxkYi = -2Im 



d^\nS{z\R',P\z,R,P) 

dX'dVi 



R'=R,P'z=P 



with X and Y standing for R or P, with the norm S being 



(3.20) 



(3.21) 



(3.22) 



5 = S{z'\R',P',z,R,P) = {z'\R!,P'\z,R,P) 
= det(/* + z'^z) 



(3.23) 



In order to implement the END-SD-FGWP formalism developed so far we need to 
specify how to construct the spin orbital basis. Usually, in electronic structure calculation, 
a set of K localized atomic orbitals are used to define the spin orbital basis. These atomic 
orbitals are given as linear contractions of Slater (STO) or Gaussian (GTO) type orbitals 
and are, in general, centered on the nuclei. These orbitals are also, in general, non- 
orthonormal. 



24 



Let VL be a matrix that transforms the non-orthonormal localized atomic orbital basis 



to the orthonormal spin orbital basis 

tl^ = <f>W 



(3.24) 

f W* W' \ 

(-/-• = I,) 

It should be noted that the bullet and open circle notation lose their meaning of occupied 
and virtual in the atomic basis. However, we are still using occupied and virtual for 
bullet and open circle even in the atomic basis representation. Using the fact that the 
only transformation that will have any effect in a determinantal state is the one that mixes 
unoccupied orbitals into the occupied orbitals, we can have that W" = 0. This means 
that the reference spin orbitals 




= (f>*W' + <I>°W° 



(3.25) 



are constructed by first orthonormalizing the bullet (occupied) atomic orbitals among 
themselves, and then orthonormalizing the open circle (unoccupied) atomic orbitals to 
the bullet (occupied) space and among themselves. This has the effect that the occupied 
(bullet) space is the same whether defined in the atomic or the spin orbital basis. That 
is the reason why we keep using occupied and virtual for the bullet and open circle 
spaces even in the atomic basis representation. We have then created a recipe of how 
to construct the reference state from the atomic orbitals basis. The construction of the 
spin orbitals involves what is called, in electronic structure theory, the MO (molecular 



25 



orbital) transformation. This is a very demanding step in terms of computational effort, 
since it scales as > K^, and the atomic integral calculation scales as w K^. As a result, 
despite the fact that the equations for the metric and dynamical variables get a little bit 
more complicated, it is of great computational advantage to work in the atomic basis. 

The spin orbitals that are delocalized over the entire molecular system are also called 

molecular orbitals. We denote the set of parameters of the coherent state with respect to 

the reference (spin orbital, or molecular) basis as and with respect to the atomic 

basis. The following transformations exist 

2“* = -I- 

( 3 . 26 ) 

Z^nol ^ 

and also the basis field operators transform as 

b = wU 



( 3 . 27 ) 




where {a, a’} are the field operators in the atomic basis representation. 
The coherent state is thus equal to 



N 



K 






t=l 



>=^ + 1 



N 



K 



= q;“* n 



«=1 



j=N+l 



( 3 . 28 ) 




at\at 



= 1 ^.-) 



26 



and as before, we work with the unnormalized coherent state 

k“r=nK+ E “pji 

»=i \ 

Note that because the anticommutation relations among are not canonical^ it is 

not possible to write the coherent state as an exponential with the atomic parameters and 
However, it is still possible to express the dynamical orbitals 

K 

Xi^^i+ 4>jZji, (l<i<N) (3.31) 

j=N+l 



^ I vac) (3.29) 



N 

Xj = - XZ 4-V’.^ (A" + 1 < i < K) 

1=1 

in the same simple form 

K 

Xi = ^i+ X] (1 < * < N) 

j=N+l 



(3.32) 



(3.33) 



where 



N 

Xj = 4>}~Y. {N + l<j< K) 

1=1 

V = - (A* -I- A'z)"^ 

= - (a* + A'^) (a' + 



(3.34) 



(3.35) 



1. the creation and annihilation field operators in the spin orbital basis are canonical since 
[6, 6^].^. = 1, however, in the atomic basis [a, = A the anticommutation is not canonical, 

since the atomic overlap matrix A has elements 

A.J = {4>i\<l>j) = J 4>i{x)4>j{x)dx 

and {<?!>, A' being non-orthononmal makes A different of the unit matrix 



(3.30) 



27 



with the overlap being given by 



S = S{z'*,E',P\z,R,P) = {z'\R',P'\z,R,P) 



= det(A* + a' z + z'^ a" + z'^A°z) 



( 3 . 36 ) 



We now define the following matrices 






A* A' /• 
A° 




^ = A* + z^A'^ + a' z + z^A°z 



( 3 . 37 ) 



and 



A° = (t. =A» + ..A' + AV + rAV 



( 3 . 38 ) 



which naturally occur in many expressions. Note that A* and A° are dependent upon 
{z,z^} and respectively. 

The metric C can now be expressed in the atomic orbital basis as 



C.. = 



dHnS 






- = -(a-^(a' + 2^A°))^.^(a-^(a' + 2+A°)) 



nk 



( 3 . 39 ) 



c...= 



d'^lnS 



= ((aV + A->)a»-> (vA' + A->))^^{A->)^„. 



and 



Cz'z' 



dHnS 



The mixed derivatives dz*V r involved in Cr are 



d 






= (a%^ + A°)a°-^(u 



( 3 . 40 ) 



^ + + <3.41) 



( 3 . 42 ) 



28 



And the double derivative with respect to the nuclear coordinates is given by 



Crr = V g^\n{z' ,Fi\z,R%i^^^R>^R 



= Tr 



A— ^(/* 






( 3 . 43 ) 






/• 



•— 1 / 






/• 

z 



where 



P-'M) 

We have introduced the notation of a vertical bar to indicate that the derivative of the 
overlap matrix is to be taken with respect to the R dependence of one side only. 

Now that explicit expressions for the metric have been given, we turn to the 
expression for the energy derivatives appearing in the right hand side of the equation 
of motion. In order to obtain explicit expressions for the derivatives of the energy it 
is appropriate to define the one-particle density matrix (also called 1-density or charge- 
bond-order matrix) as follows 



r = (^J)A-'(/* 2'). 

which was obtain from the kernel 

Tji{z*,z) = {z\zy {z\blbj\z) 
and has the familiar block form 




( 3 . 45 ) 



( 3 . 46 ) 



( 3 . 47 ) 



29 



The energy derivatives then become 
dE{z\z) 



+ /°)(/i + Tr(Va6;„6r)J 

= ^ (a'^i;+ + A°) ( u P )F ^ ^ A*" 

where we recognize the Fock operator 

F = h + Tr(yoj,;aj,r)^ 

= h-\- Tr([(aa|66) — (a6|fea)]r)^ . 

The energy derivatives with respect to the nuclear coordinates are 






ki 



1=1 



(3.48) 



(3.49) 



F{z*,R',P\z,R,P) I 

l^ZkZie^(^Rk-Ri) / \ / \ 

= -5 E + Tr(v^.Ar) - Tr(v^-,Arftr) (3.50) 



+ 5Tr(Tr(v^^Ki;,»r) j) - Tr('Tr(v^^Ary,i.,tr) J 
and since the basis set is independent of the velocity (or momentum) of the nuclei the 
energy derivatives with respect to the momenta become 



VpE{z\R',P\z,R,P) 



= fL 

R’=R,P'=P Mk 



(3.51) 



We now have explicit equations for all terms in the equations of motion 



/ iC 


0 iCR 




i 




1 


( OF !dz*\ 


0 

• 


-iC* -iC*p 


0 




z* 




dEldz 


iCn 


-iCp Crr 


-7 




R 




dEjdR 


\ 0 


0 I 


0 




\pj 




\ dEIdP / 



(3.52) 



30 



and we can try to simplify the main equations in such a way that they become efficient 
to implement in a computer program. Starting with the equation 



+ iCRR 



dE 

dz* 



( 3 . 53 ) 



we can use the expressions for Cg>z, Cr, and dEjdz* to further reduce it to 






/• 

2: 



= ( -2 1° )F 



/• 



( 3 . 54 ) 



The right hand side has a very simple interpretation: the Fock operator acts on the 
occupied states and is then projected on the virtual space. Thus only the projection on 
the virtual states gives rise to a change in the 2 coefficients. This is also the way that 
this particular equation of motion is implemented since it does not involve any matrix 
inversion, which would be necessary if the metric did not have this particular structure. 

In the equation of motion that involves the derivatives of the energy with respect to 
the nuclear coordinates, 



E{z\H',P\z,R,P) I 



= 4E 



^ ZtZ,e‘‘{Rt - Rl) 



/=! 



\Rk - Ri\^ 



+ Tr(v^-/r)-Tr(v^^Ar/.r) (3.55) 



+ ^Tv[Tr(v^Vab-,abT)r^ - Tv ( tv ( v ^ATVab.,abr) T 



31 



we can see that it will be necessary to compute the derivative of the 1-density matrix 

r= (^*j(^A* + A'z + z^A'^ + z^A°zy\r z^) 



_M 1a— !(/. 



/• 

Z 



V(/* \i' ^') 



with respect to the nuclear coordinates, which can obtained as follows 



= ; r 



)(V^.A-)(/' 









r\..- 

z 






(/• .')(v^.a)(^Qa-‘(/' ^') 






(3.56) 



(3.57) 



= r(v^,A)r. 

As a result, in order to compute the energy gradient with respect to the nuclear coordinates 
all we need are the gradients of the one- and two-electron integrals (which are available 
from many quantum chemical program package) in addition to the derivatives of the 
atomic overlap matrix. 

The NDDO Approximation 



The NDDO (neglect of diatomic differential overlap) approach employs the following 



approximations [4]: 



32 



1. Only the valence electrons are considered explicitly in the calculations. The nucleus 
and inner shell electrons are replaced by a fixed core function. This is the so called 
core approximation [44]. 

2. A minimum basis set is used. This basis set is constructed with Slater type orbitals 

(STO’s). The advantages of using STO’s will be clarified later. The STO’s are 
products of a radial function and a normalized real spherical harmonics 

with quantum numbers n, /, m, 

Xnlm = Rnl{r)Yi^{e,(p), 



(or ,'\n+l/2 (3.58) 

R ,(r) = -n-1 -Cnir 

' [(2n)!]V2 ' • 

Here Cn/ is the orbital exponent of the Slater atomic orbital. 

3. The overlap integrals are neglected, except when they appear in the one-electron 
resonance integrals (one-electron two-center integrals). As a result, the NDDO 
approximation consists of applying the following relation 



(1) ^ SabX^{1)x^{1) . (3.59) 



Since we are treating only valence electrons and using STO’s we have that 




(3.60) 



the atomic overlap matrix is approximated by the identity matrix. 



4. Two-electron integrals then become 



^^C<Bc(#^(l)tf(2)|^^(l)^f(2)) (3.61) 



33 



We have chosen the NDDO model [45, 4] since it is considered the lowest level 
at which there exist a basis set for which the zero differential overlap (ZDO) [46] 
approximation is valid, or almost valid [47, 48]. The NDDO model is rotationally 
invariant so that it is not necessary to perform any spherical averaging of the atomic 
orbitals with angular momentum larger than zero. As a result, unlike the other ZDO 
models, such as CNDO (complete neglect of differential overlap) and INDO (intermediate 
neglect of differential overlap), the NDDO model includes orbital anisotropies. In 
addition, in the HF-NDDO method the computational dominant step is the diagonalization 
of the Fock matrix. 

We now present some justifications for the approximations adopted and their advan- 
tages: 

(1) The core approximation in the NDDO model is reasonable because the inner 
electrons are tighdy bound and are, therefore, unlikely to be significantly perturbed by 
changes in the valence shell. Consequently, most chemical processes can be accurately 
described by valence electrons only. Also, there are solid theoretical foundations for the 
core approximation in terms of an effective Hamiltonian [49]. 

(2) The use of STO’s as building blocks for the molecular orbitals ensures that only 
a few of these STO’s are necessary to give a good description of neutral molecules 
or positive ions. This justifies the approximation of minimal basis set. Only in the 
cases of anions or hypervalent compounds must the atomic basis set be more flexible to 
accommodate the extra charge density or extra bond(s). Adding diffuse or polarization 
atomic orbitals is one way of solving this problem. Another is to allow the orbital 



34 



exponents to vary with the atomic charge. This provides an appropriate description for 
molecular anions [50], and also allows us to keep the minimum basis set formalism. The 
use of STO’s is useful in the gradient calculation. That is, the differentiation of an STO 
function with respect to the nuclear coordinate does not yield higher angular momentum 
functions. In other words, once the basis set is defined in terms of STO’s no higher 
angular momentum functions are needed in the gradients of the molecular integrals. 

(3) The neglect of overlap is consistent with the approximation employed in the two- 
electron integrals. However, keeping the overlap in the one-electron resonance integrals 
(one-electron two-center integrals) is important since these integrals are responsible for 
the bonding in molecules. That is, the resonance integrals 

= (x„(l)l(-iv?-^:?i)|x.(l)), (3-62) 

which involve an overlap distribution are not neglected. 

(4) It can be shown [51, 52] that the approximation of the two-electron integrals by 
the neglect of diatomic differential orbital is correct up to second order in the overlap 
expansion. It is also possible to evaluate the three- and four-center integrals in terms 
of overlap and two-center integrals [52]. However, if we parametrize the two-electron 
integrals with respect to experimental data the errors caused by neglecting the three- and 
four-center integrals can be minimized. 

The END-SD(NDDO)-FGWP Method 



Since the dynamical equations are dependent upon the norm we start by applying 



35 



the NDDO approximation to 



S = det(A* + A' z + z'^A" -f z'^A^z). 
The NDDO approximation implies that 







A* = r, A' = A" = o, A° = r 

so that the overlap takes the same form as in the orthonormal basis, 

S = det(7* + z'^z) 



as do the A matrices 

A* = /• + zh 



A° = 7° + zzK 



Also, the density matrix is simplified to 



r= 1^*1 A— ^(7* z^) = 



(^'’yr + z'z)-'{r zt) 



(3.63) 



(3.64) 



(3.65) 



(3.66) 



(3.67) 



and 

u = -(A'^ + A°2)(A* + A'z)-^ = -z 



= -(A* + z^A'^)-^(A' + z^A®) = -z+ 

As a result, the relevant terms for the dynamical equations become 

Metric 




d'^lTiS 



-((r + z*z)-'z^).,{{r + z<z)-'z\, 



(3.68) 



(3.69) 



36 






(3.70) 



Im ^ki 



(3.71) 



d 






= (/° + ^zt)-l(_^ /'>)V^-^A|i: (/* + zM 




K\-i 



Crr = V ^\n{z' ,Ii\z,R)\^'^^^Il-^R 



(3.72) 



= Tr 



(/• + 2^2)-^(/* 



V^AI ^ 




-(r + z'z)-'(r z^)v^A\(''){r + z'z)-\r z «) v ^ a /^‘ 




Energy Gradients 



dE{z\z) 



8z* 



= ((/° + 22')-‘(-z r)F{’'\r + zU)-' 



ki 



V^E{z\B!,P\z,R,P)\^ 

"* \R'=R,P>= 



^ZkZ,e^(^Rk-Ri) 



1 

=—y 

2 ^ 



1=1 

l^k 



- RiP 



+ Tr(Vjj,Ar)-Tr( 



V^AThr) 



+ 5Tr('Tr(v^^Ki..ir) - Tr(Tr(v^^ArK.,,.ir) F 



(3.73) 



(3.74) 



(3.75) 



Vpy{z\R',P\z,R,P) 



= A 

R'=R,P'=P Mk 



(3.76) 



37 



where the Fock operator must also be considered under the NDDO approximation 

f = A + Tr(v;^S'>°r)^ 

(3.77) 

= h + TT{[{aAaA\bBbB) ~ {aAbA\bBaB)]^)a • 

Caution must be exercised when considering the derivatives of the atomic orbital 
overlap A with respect to the nuclear position. If we make the NDDO approximation 
for the overlap, that is, A = /, then take the derivative we obtain no contribution. 
As a result, both Cr and Crr vanish and we do not have any coupling between the 
electrons and the nuclei. This is not consistent with the ab initio END equations, since the 
NDDO approximation should not affect the proper coupling between electrons and nuclei. 
Also, making the NDDO approximation before taking the derivatives is inconsistent with 
the analytical gradient method. That is, the contribution of the two-center one-electron 
(resonance) integrals to the energy gradient would then vanish, which is not true when 
finite difference method is employed. Consequently, we should first take the derivative of 
the non-approximated atomic orbital overlap and then perform the NDDO approximation 
on the resulting expression. 

AMI Model 

The AMI (Austin Model 1) approach to the NDDO approximation [53] has been 
extensively tested and has been shown to reproduce geometry, heats of formation, dipole 
moments, and ionization potentials very well [53-55]. In addition, the overall perfor- 
mance of the AMI for computing molecular properties (excitation energies, vibrational 
frequencies and intensities) and quantities relevant for chemical reactions (enthalpy of 



38 



reaction and activation energies) is also satisfactory [55]. It, thus, seems appropriate to 
use the AMI realizations of the NDDO method in our END-SD(NDDO)-FGWP model. 
Results for dynamical properties like intra- and intermolecular charge transfer, transition 
state spectroscopy, photodissociation, etc, can be used to check the performance of the 
END- AMI method. In case this approach fails a different realization of the NDDO 
method can implemented where the parametrization should also include experimental 
dynamical properties in the data base. 

Before we present the AMI model we need to consider first its Fock matrix elements 
[56, 53, 55]: 

1 . Diagonal terms 



R 






= u, 



(tA/iA 



Bi^A 




(3.78) 






B^A Xb 



2. Off-diagonal terms on the same atom 







Bi^A 



+ l:P^^Al'A[^y'Al'A\^^AVA) - {tiAiiA\vAt'A)\ 



(3.79) 



+ '^'Y^P\B<TBit^AVA\^B(^B)- 



B^A Xb ^b 



39 



3. General terms on different atoms 




I'A <TB 



where <^b) are atomic orbitals centered in A {B) and P^x is the bond 

order matrix. 

In order to complete the definition of the AMI model we need expressions or numbers 
for each one of the terms that go into the elements of the Fock matrix. That is, 

!• UfiAHA one-electron one-center terms and are parametrized from spectroscopic 
data for valence states of atom A and its ions; 

2. Za is the atomic number of atom A minus the number of core electrons; 

3. The nuclear repulsion energy is given by 



where af, bf, and cf are parameters; 

4. The resonance integral is made proportional to the atomic orbital overlap. The 
proportionality constant is the average of the parameters; 



Eab = ZaZb{sasa\sbsb) 




4 



(3.81) 



5. The repulsion integrals are expanded in terms of multipole Mi^ and are approximated 



40 



by the following expression 

00 OO imtn 

/ l =0 / 2— 0 



where, 




Rij = ^AB + + vf,D^ (3.83) 

with tj = ...,±2,±1,0 depending upon the order of the dipole, p and D are derived 
parameters, that is, D = /^(C/i) and p = p{U,i„,D). In the case of a minimum sp 
basis set, there are 22 distinct repulsion integrals. The inclusion of d orbitals increase 
this number to 450, which make this type of approximation for the repulsion integrals 
cumbersome to be extended beyond sp basis. 



The number of parameters in the AMI model ranges from 13 to 16. Around 150 
molecules were included in the reference data base used to obtain those parameters. These 
numbers can give an idea of how difficult is to obtain parameters for a semiempirical 
method. 

Implementation 

The END-SD-FGWP method has been implemented in the ENDyne program package. 
This program is capable of performing simultaneous optimization of the electronic and 
nuclear (geometry, for classical nuclei) wave functions. ENDyne also can perform the 
time evolution of a molecular system state. The input consists of the initial states 



41 



(electronic and nuclear), number and type of particles (electrons, nuclei, nuclear masses, 
atomic number), initial and final time of the evolution, the (Gaussian) basis set for each 
nucleus, and the type of differential equation solver. The program works in the laboratory 
frame and uses atomic units. As output, ENDyne generates a file, called restart which 
contains the history (the 2 parameters and phase) of the system evolving in time. Another 
program, called EVOLVE, is used to extract, plot, and manipulate relevant information 
like nuclear position, electronic and nuclear momenta, phase, electron density and atomic 
population, etc, from the restart file. 

A functional diagram of the ENDyne program can be seen in the Figure 3-1. 






42 




Figure 3-1: Functional diagram of the ENDyne program. 



The main program endyne calls getinp and decides the structure and type of calcula- 
tion. After that, it makes a dummy run and determines exactly how much memory will be 
needed for the calculation. Then, the rundyn is called and it takes control of the ENDyne 
calculation. It calls dynini to initialize the calculation for the models (quartic, Morse, 
and oscillator potentials) and theories (Hartree-Fock, AGP, RPA). It then decides if the 



43 



calculation is an optimization (frprnm, djpmin), optimization (Jrprmn, dfpmin) followed 
by evolution {deint, drint, odeint), only evolution(dc, driveb, odeint), or exact oscillator 
(xctosc). Dynfun is then called and computes the gradient of the energy with respect to 
the wave function parameters in the case of optimization or the derivatives of all wave 
function parameters with respect to time for an evolution calculation. The appropriate 
forces (gradients) for the semiempirical or ab initio model is then called. In the case 
of the Hartree-Fock theory (END-SG-FGWP) the atomic integrals and overlap and their 
derivatives are computed every time the force routine {hffrc) is called. The onedrv and 
twoint routines are drivers for one- and two-electron integral calculations, with averll, 
averll, and disint being the interfaces with the ABACUS integral package. 

In the case of AMI model the structure is the same, except that the drivers onedrv 
and twoint calls some other appropriate interfaces, instead of calling averll, averll, 
and disint. In addition, the implementation of the AMI method, necessitates the 
development of equations and source code for computing analytical derivatives of the 
integrals (repulsion and overlap) and of the energy. The molecular orbital programs 
AMPAC 2.1 and MOPAC 6.0 based upon the AMI model do not perform analytical 
derivatives at the SCF (self-consistent field) level. However, analytical derivatives for 
similar expressions of the repulsion integrals have already been performed [57], which 
insures that the AMI model implementation is viable. 



CHAPTER 4 

H+ + H, He, AND H 2 CHARGE TRANSFER COLLISION 



In this Chapter we present results for the electron-nuclear dynamics that takes place 
when H"^ collides with H, He, or H 2 . 

Such a study consists of computing the probability of the charge transfer, energy 
transfer, etc as a function of the impact parameter and, in the case of the H 2 target, also 
molecular orientation. Plots of the transfer probability, vibrational excitation, and cross 
sections versus the scattering angle are presented and comparisons with other theoretical 
approaches as well as experimental data discussed. 

This chapter has the following subsections: the H'*' + H, + He, and + H 2 results, with 
a fourth subsection containing general conclusions. 

Before presenting the results some general considerations are in order. 

i) All the calculations were performed with the ENDyne program package using the 
following computers: IBM RS/6000-550, Sun 4/490, Sun 690MP, and Cray Y-MP/432. 

ii) An Is2s2p basis set was used for both H and He. This basis sets, namely pVDZ 
for H: [4slp]/(2slp) [58] and 6-31G* for He: [4slp]/(2slp) [59] are listed in Table 4-1. 
In the pVDZ basis set the s-exponents were scaled by 1.2^ = 1.44, as was done in the 
original DZP basis set by Dunning [60]. 

iii) Some of physical properties for the atoms and molecules involved in this work are 
presented in Table 4-2. 

iv) Due to the physics of the problem (a proton hitting atomic and molecular targets) the 



44 



45 



initial wave function consisted of a totally localized electron density on the H, He, and 
H 2 targets. In other word, the starting wave functions are not the ground states of the 
complex species (H-H)"^, (H-He)"^, or (H-H 2 )‘^. It should be mentioned, however, that 
the atomic and molecular targets are in their electronic (and vibrational) ground states. 
The difference in energy associated with these initial wave functions and the ground 
state energy of the (super)molecule ion species is due to the localization of the electronic 
density around the targets. 



Table 4-1: Contraction coefficients c and exponents a for the basis sets used in this work. 





H/pVDZ“ 


He/6-3 




c 


a 


c 


a 


Is orbital 


0.01969 


18.7344 


0.02311 


38.4216 




0.13798 


2.82528 


0.15468 


5.77803 




0.47815 


0.64022 


0.46930 


1.24177 


2s orbital 


0.50124 


0.17568 


0.29796 


1.00000 


2p orbital 


1.00000 


0.72700 


1.10000 


1.00000 



a: Ref. [58], b: Ref. [59]. 



Table 4-2: Ionization potentials, electron affinities and polarizabilities of H, He and H 2 . 





Ionization Potential 
(eV) 


Electron Affitinity 
(eV) 


Polarizability 
(lO'^"* cm^) 


H* 


13.598 


0.754 


0.6668 


He® 


24.587 


not stable 


0.2050 


H 2 ® 


15.427 


— 


0.803 



a: Ref. [61]. 



46 



The + H Collision 

The study of H'*' + H collisions is of special interest because of its great simplicity 
but rich physics. This has inspired much theoretical and experimental work involving 
charge transfer (total and differential cross sections), excitation, alignment measurements, 
etc. As a result, H'*’ + H collisions provide a good test for any time-dependent method 
that intends to describe electron and nuclear dynamics. It should be noted that since this 
is a one electron system the effects of electronic correlation are absent. Consequently, 
the same methods should be tested for systems with more than one electron, since like 
in stationary quantum chemistry, the electronic correlation effects can be essential when 
describing molecular properties. 

Most of the theoretical calculations on the H'*’ + H system are done by close- 
coupling methods [25, 24]. Since these methods use prescribed trajectories they are 
most commonly applied to collision energies above 1 keV in the laboratory frame. One 
of the investigations in this chapter includes the comparison of dynamical trajectories 
obtained with the END-SD-FGWP method with prescribed Coulomb trajectories [34]. 
Below 1 keV there are just a few theoretical studies where the PSS (perturbed stationary 
states) method is used to separate the nuclear and electronic degrees of freedom [62]. 

Total Cross Sections 

The scattering of a proton by a hydrogen atom has already been studied by the 
END-SD-FGWP method [38]. In that work an Is2s2p basis set was also employed but 
each function was expressed in terms of six primitive Gaussian functions. Since in this 



47 



work we are using a smaller number of primitives we decided to compute the total cross 
sections at several energies and compare them with experimental and previous END 
work. This comparison should provide us with some idea of how the basis set affects 
the total cross section. 

We have used the following initial conditions: i) the hydrogen atom is in its ground 
state with the following coordinates (20.0, 0.0, 0.0) in a.u. with respect to the laboratory 
frame; ii) the proton is placed at the (0.0, b, 0.0) coordinates, with b (the impact parameter) 
varying from 0.0 to 15.0 a.u.; iii) the hydrogen atom has zero momentum; iv) the proton 
has initial momentum corresponding to the energy of the collision; v) the time evolution 
of the system is carried out until the nuclei are separated by more than 1(X) a.u. The 
transition probability is obtained by projecting the asymptotic state of the proton on 
the eigenstates of the moving proton for elastic processes or hydrogen atom for charge 
exchange [38]. 

Total cross sections of the scattering of H'*’ on H are obtained by numerically 
integrating [63, 64] 

CX) 

<7{E) = 27t / bP{b;E)db (4.1) 

0 

for several energies E (0.5, 1.5, 5.0 keV, e.g.), where b is the impact parameter and 
P{b] E) the transfer probability as a function of P(6; E) for a given E. The upper 
integration limit is chosen to be between 11.0 and 12.0 a.u.. This choice is reasonable, 
since the transfer probability in this interval is smaller than 10~^. 

For a collision energy of 0.5 keV, we obtain the results summarized in the Table 
4-3, where also the total cross section is shown as calculated using different integration 



48 



steps (increments in the impact parameter). As we can notice from Table 4-3, the best 
compromise between intervals in the impact parameter and the cross section accuracy 
is the last row, where we have an increment of 0.10 a.u. in the oscillatory region (0.0 
- 6.0 a.u.) and 0.50 a.u. in the monotonically decreasing region (6.0 - 11.0 a.u.), see 
Figure 4-2 for visualization. 



Table 4-3: Total cross section for + H collision at 500 eV using pVDZ basis set. 



Interval (a.u.) 


Increment in b 
(a.u.) 


Interval (a.u.) 


Increment in b 
(a.u.) 


Cross Section 
([a.u.]2) 


0.0 - 11.0 


0.05 






64.0583 


O 

• 

1 

O 

• 

o 


0.10 






64.0583 


0.0 - 11.0 


0.15 






64.0671 


0.0 - 6.0 


0.05 


6.0 - 11.0 


0.15 


64.0589 


0.0 - 6.0 


0.10 


6.0 - 11.0 


0.50 


64.0521 



49 




Impact Parameter (a.u.) 

Figure 4-2: Weighted transition probabilities for total electron transfer at 500 eV as a function 
of impact parameter from ENDyne. All data in atomic units. 

Table 4-4 summarizes the total cross section results obtained by the END-SD-FGWP 
method using two different sets of primitive functions in the basis sets. 



Table 4-4: Total transfer cross sections for H* colliding with a H (xlO"*^ cm^). 



Total Transfer Cross-section. 


Collision 


END-SD-FGWP Theory 


Experiment^*’ 


Energy (eV) 


pVDZ basis 


STO-6G basis 


10 


35.93 


36.37 


37.0+5.7 


100 


24.31 


25.60 


23.7+3.5 


500 


17.94 


19.44 


18.9+3.2 


1000 


16.55 


16.78 


16.3±2.9 



a: Newman et ai. Ref. [65], b: Gealy and Van Zyl, Ref. [66]. 



50 



As we can see from Table 4-4 the basis set do not have a large effect in the calculated 
integral cross section for electron transfer in the + H collision. In addition, as noted 
in early studies the END-SD-FGWP method provides excellent results for charge transfer 
cross sections when compared with experimental data. 

Differential Cross Sections 

Since the integral cross section involves an integration much of the detailed behavior 
of a charge exchange collision is masked. Such details can be obtained from state-to- 
state cross section, differential cross section, and transfer probability as a function of the 
scattering angle and consequently should be the basis for theoretical model analysis. We 
present results for the transfer probability and the reduced differential cross section as 
a function of the scattering angle for the H'*' colliding with H at several energies (0.25, 
0.41, 0.50, 0.70, 1.0 keV). 



The expression for the differential cross section (aj) at a given collision energy E 



is [63, 64] 



CTd = 



da{e) b P{b) 



dO, 



sin 9 



d6 

IE 



(4.2) 



where b, 6, P{b), dil = sin 0 dO d(f> are the impact parameter, the scattering angle, the 
transfer probability, and the solid angle, respectively. The modulus of db/d6 is used to 
insure that the differential cross section is always a positive quantity. In addition, since 
the scattering angle by definition always lies between zero and tt, we have that sin 0 > 0. 
It should be noted that the differential cross section depends upon the derivative and is 



not integrated over any variable so that more details of the collision process are revealed. 



51 



We should now make the connection between the classical deflection function 0 and 
the scattering angle d. The classical deflection function 0 is given by [63, 64] 




where V(r) is the central potential, E the collision energy, and the distance of closest 
approach a is obtained by solving the equation 




(4.4) 



From the expression for the deflection function we can see that 0 may have any value 
between — oo and x, and as we mentioned before 0 < 0 < tt. As a result, the two angles 
have to be connected by 



Q = TjO — 2nv 




where t; = ±1 and n is a positive integer (or zero). For a given 0, the numbers t) and 
n are chosen so that 6 lies between zero and tt. We can also show that 



de dd 


de 




de 


db ~"^db ^ 


db 


— 


db 



(4.6) 



and consequently. 




(4.7) 



52 



Trajectory Analysis 



As mentioned before time-dependent methods using prescribed trajectories are not 
appropriate for calculating the differential cross sections since they involve the derivative 
of the deflection function 0 with respect to the impact parameter b. The END-SD-FGWP 
method with its dynamical trajectories should be more suitable. We present a comparison 
of 0 versus 6 using bare nuclei and a hydrogen atom potential. The central potential in 
the case of bare nuclei has the following form 

r) = z= r~^ (4.8) 

r 




for two protons. The resulting analytical solution for the relation between impact 
parameter and scattering angle is the so-called Rutherford scattering formula 




This is certainly not a realistic interaction potential between a proton and a hydrogen 
atom. Instead, using the exact wave function for the ground state of a hydrogen atom 
we obtain the following potential for the electrostatic interaction between a proton and 
a hydrogen atom in the ground state [67] 



I/(r) = (l-|-r-^)e-2". 



(4.10) 



There is no analytical solution of the classical deflection function for this potential, so 
we have to solve it numerically and the results are plotted in Figure 4-3. 



53 




Figure 4-3: Scattering angle as a function of impact parameter using ENDyne, bare nuclei, and 
hydrogen atom potential for the + H system. Eneigy: 500 eV. Basis set: pVDZ. Scattering 
angle in degrees and impact parameter in a.u.. Full line: ENDyne, dotted line: bare nuclei, and 
dashed line: hydrogen atom potential. 

A more detailed illustration of the differences between these three trajectories can 
be seen in Figure 4-4. It is clear from Figure 4-4 that the errors due to prescribed 
trajectories can be quite large for small impact parameter. We also notice that the 
dynamical trajectory lies between the bare nuclear and the hydrogen atom potentials. 
This is due to the fact that in the H'*’ + H system charge transfer is expected so the 
interaction potential should be represented by a combination of the bare nuclei and the 
atomic potential. The contribution of each extreme potential (bare nuclei and atomic) 
is related to the amount of charge transfer during the collision. Consequently, the use 
of prescribed trajectories in calculating properties that depends upon the scattering angle 
becomes doubtful for energies below 1 keV and for small impact parameters. 



54 




Figure 4-4: Scattering angle as a function of impact parameter using ENDyne, bare nuclei, and 
hydrogen atom potential for + H system. Small impact parameter region. Energy: 500 eV. 
Basis set: pVDZ. Scattering angle in degrees and impact parameter in a.u.. Full line: ENDyne, 
dotted line: bare nuclei, and dashed line: hydrogen atom potential. 

Transfer Probability 



As we can see from the expression for the differential cross section, equation (4.7), 
the transfer probability is an important contributing factor for this quantity. We then 
present in this subsection END-SD-FGWP results for this probability as a funciton of the 
scattering angle and compare them with the experimental data. 

Before performing these comparisons we present a brief discussion of the experi- 
mental procedure and try to identify the main sources of experimental uncertainties. The 
experimental procedure consists of [68, 69] a mass analyzed H"'' beam issued from a 
single discharge ion source crossed with a thermal H beam. Ions and atoms scattered 



55 



at a given angle are selected and analyzed. The overall energy resolution varies from 
0.8 eV up to 2 eV in the 250 eV — 2000 eV energy range. The uncertainties affecting 
the experimental results arise from i) the finite angular resolution AO (± 0.07° for small 
angles up to ± 0.2° for large angles), ii) the errors in the dissociation fraction determina- 
tion (H/H 2 ratio in the collision region), and iii) the accuracy of the relative calibrations 
of the cross sections for the different processes (elastic, charge transfer, and excitation). 
Because the rapid variation of the transfer probability and the differential cross section 
with the angle, more particles arise from the region 6 — AO than from the region 0 -f- A^. 
Thus the center angle is slightly larger than the one corresponding to the most prob- 
able scattering. However, this error never exceeds 0.1° [68, 69]. As we shall see in 
the comparison between the experimental data and the END-SD-FGWP results the finite 
angular resolution can account for most of the damping in the oscillatory behavior of the 
observed versus calculated transfer probability and differential cross sections. Problems 
with the measured direction of the incident beam with respect to the detection system 
can be eliminated by studying the scattering on both sides of the incident beam direction 
and choosing the zero index from the symmetry of the data curves. In addition to these 
uncertainties, the experimental results are normalized to the theoretical results [70] for 
the elastic differential cross section at small angles. 

From the discussion of the experimental uncertainties it seems clear that the measured 
transfer probability versus the scattering angle is less susceptible to errors than the 
differential cross section and they are independent of scaling factor. Thus in Figures 
4-5-4-9 we present the results for the transfer probability calculated by the ENDyne 



56 



program and compare with the experimental data. The theoretical procedure is the 
same as described in the calculation of the integral cross section. The only additional 
information needed is the asymptotic momenta of the proton, which are used to compute 
the scattering angle, that is 

Px Px 1 / .1 1 1 \ 

COS0 = = . , = = (4.11) 

Ptot y/Px+Py y/1 -f {PylpxY 

where px and py are the x- and y- components of the asymptotic proton momentum. 




Scattering Angle (deg) 

Figure 4-5: Transfer probability versus the scattering angle (in degrees). Comparison between 
the experimental and ENDyne results for the H* + H system. Energy: 250 eV. Basis set: pVDZ. 
Experimental angtilar resolution ±0.6°, ’+’ ENDyne, from Ref. [68], and ’x’ from Ref. [69]. 



57 




Figure 4-6: Transfer probability versus the scattering angle (in degrees). Comparison between 
the experimental and ENDyne results for the + H system. Energy: 410 eV. Basis set: pVDZ. 
Experimental angular resolution ±0.2° — ±0.6°, '+' ENDyne, firom Ref. [68], and ’x’ from 
Ref. [69]. 




Figure 4-7: Transfer probability versus the scattering angle (in degrees) for the H* + H system. 
Energy: 500 eV. Basis set: pVDZ. ’+’ ENDyne. 



58 




Figure 4-8: Transfer probability versus the scattering angle (in degrees). Comparison between 
the experimental and ENDyne results for the + H system. Energy: 700 eV. Basis set: pVDZ. 
Experimental angular resolution ±0.02° — ±0.6°, ’+’ ENDyne and from Ref. [68]. 




Figure 4-9: Transfer probability versus the scattering angle (in degrees). Comparison between 
the experimental and ENDyne results for the H"^ + H system. Energy: 1000 eV. Basis set: pVDZ. 
Experimental angular resolution ±0.07° — ±0.6°, ’+’ ENDyne and from Ref. [68]. 



59 



Both the qualitative and quantitative features (position of maxima and minima) of 
the experimental data are very well reproduced by the END-SD-FGWP method The 
quantitative agreement can be made much better if we apply an angular resolution window 
to the theoretical data. This would simulate the damping in the experimental data and 
explain why the oscillations in the transfer probability do not extend form zero to unity. 
There is still some discrepancies, mainly at high energies and large angles, between our 
theoretical treatment and the experimental data. We believe that the extra features in 
the theoretical results are real and were not picked up by the experiment due to the poor 
angular resolution. It also should be mentioned that the semiclassical treatment [70] using 
a three-term molecular basis provides results in perfect agreement with the experimental 
data. This is puzzling since the experimental data have a damping of the oscillation 
due to the finite angular resolution and the theoretical work does not mention how this 
damping is handled. Because of this the quantitative agreement between the semiclassical 
method and the experimental results seems to be fortuitous or the true meaning of the 
theoretical data is difficult to understand. We should add that this damping is also not 
observed in other theoretical treatments [34]. 

Reduced Differential Cross Section 



The experimental results available for the H'*’ + H collision are often presented as 
’reduced’ differential cross sections (p) which are expressed as [68] 



P 





be P{b) 



db 



(4.12) 



60 



The reason for this is due to the rapid variation of the differential cross sections with 
the scattering angle. So, in the following Figures (4-10-4-14) we present both elastic 
and charge transfer reduced cross sections calculated using the ENDyne program and 
compare them with the experimental data. 




0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 



Scattering Angle, 0 (deg) 

Figure 4-10: Reduced differential cross sections versus the scattering angle. Experimental and 
the ENDyne results for the H"^ -h H system. Eneigy: 250 eV. Basis set: pVDZ, ENDyne 
transfer, ENDyne elastic, ’x’ transfer and ’o’ elastic from Ref. [68]. 



61 




Figure 4-11: Reduced differential cross sections versus the scattering angle. Experimental and 
the ENDyne results for the -i- H system. Energy: 410 eV. Basis set: pVDZ, ENDyne 
transfer, ’+’ ENDyne elastic, ’x’ transfer and ’o’ elastic from Ref. [68]. 




Scattering Angle, 0 (deg) 

Figure 4-12: Reduced differential cross sections versus the scattering angle. Experimental and 
the ENDyne results for the H system. Energy: 500 eV. Basis set: pVDZ, ’*’ ENDyne 
transfer, ’+’ ENDyne elastic, ’x’ transfer and ’o’ elastic from Ref. [68]. 



62 




Figure 4-13: Reduced differential cross sections versus the scattering angle. Experimental and 
the ENDyne results for the + H system. Eneigy: 700 eV. Basis set: pVDZ, ENDyne 
transfer, ’+’ ENDyne elastic, ’x’ transfer and ’o’ elastic from Ref. [68]. 




Scattering Angle, 0 (deg) 

Figure 4-14: Reduced differential cross sections versus the scattering angle. Experimental and 
the ENDyne results for the H* + H system. Energy: 1000 eV. Basis set: pVDZ. ’*’ ENDyne 
transfer, ’+’ ENDyne elastic, ’x’ transfer and ’o’ elastic from Ref. [68]. 



63 



The same comments and conclusions about the transfer probability hold for the 
reduced differential cross sections. We should only add that the agreement between 
the END-SD-FGWP and experimental results for the reduced cross sections is worse 
than for the transfer probability. The reduced differential cross sections are explicitly 
dependent upon the scattering angle. The experimental data were normalized and fitted 
to theoretical results, which are based upon prescribed trajectories. We can then argue 
that the experimental results for the reduced cross sections have been downgraded in 
quality by fitting to unsuitable theoretical results. 

The H~^ + He Collision 

The behavior of proton-helium scattering is different from that of the proton-hydrogen 
collisions. The -f He system exhibits electron-electron interaction, the charge transfer 
is non-resonant, and the interaction potential has repulsive and attractive parts. These 
facts make the scattering of a proton by a helium atom an important system to study 
with any new time-dependent method that has demonstrated the ability to describe well 
the -t- H collision. 

Since the early 1930’s the proton-helium scattering has been studied by theoretical 
methods [71]. The accumulated theoretical data falls mostly in the energy range above 10 
keV and are related mainly to total (integral) cross sections. Experimental and theoretical 
studies of differential cross sections for + He are more scarce. However, it is possible 
to find several experimental studies for energies down to 5 keV and scattering angles 
greater than 1° [72]. These experiments are probing the repulsive part of the interaction 
potential and no structures are then observed. Recent studies of angular differential 



64 



scattering at keV energies in ion-atom collisions have focused on very small angles, 
in general, below 1° [73]. These studies are motivated by the highly forward-peaked 
character of the cross sections and by the location of the classical rainbow angle within 
the 0-1° range (at keV energies). The fact that structures occur for very small angle 
has been the main reason for the delay in investigating this region, since high angular 
resolution is needed. From the theoretical point of view, the oscillatory patterns seen in 
elastic and inelastic scattering have been studied extensively using semiclassical scattering 
theories [63, 74, 75]. 

Trajectory Analysis 



Some of the dynamical trajectories obtained from an ENDyne calculation of a proton 
being scattered by a helium atom, with collision energy of 50.0 eV is presented in Figure 
4-15. In general, these trajectories should be expected in any ion-atom collision non- 
resonant scattering. 



65 




Figure 4-15: Dynamical trajectories of being scattered by He. ENDyne results at energy of 
50.0 eV with a pVDZ basis set. 

The appearance of classical rainbow scattering implies that the classical deflection 
function has at least one extremum. This means that there are at least two impact 
parameters that will yield the same scattering angle. In the case of a proton colliding 
with a helium atom we show in Figure 4-16 an example of three dynamical trajectories 
that yield the same scattering angle. 



66 




Figure 4-16: Dynamical trajectories for scattering angle of 2.65°. ENDyne results for + He 
system at 50.0 eV with a pVDZ basis set. Impact parameters: solid line ( — ) =1.12 a.u., dashed 
line ( — ) = 1.60 a.u., dotted line (• • •) = 2.10 a.u. 

We should point out that apparently the target (He) moves only in the perpendicular 
direction (y-axis) to the scattering axis (x-axis). However, this is due to the scale of the 
graphic. In fact, the atomic target is first attracted to the incoming ionic projectile and 
then repelled, as can be seen in Figure 4-17 for three selected impact parameters. 



67 




Figure 4-17: Motion of the atomic target (He) during a collision with at 50.0 eV as computed 
by ENDyne with a pVDZ basis set. The initial coordinate of the He atom is (15.0, 0.0). Impact 
parameters: solid line ( — ) = 1.12 a.u., dashed line ( — ) = 1.60 a.u., dotted line (•••) = 2.10 a.u. 

The Deflection Function and the Interaction Potential 

In general, the deflection function of an ion-atom system has at least one extreme 
point. The exception is the resonant collision of a proton with a hydrogen atom. This 
behavior of the deflection function is due to the interaction potential which for the non- 
resonant case has both repulsive and attractive parts. We present a detailed analysis of 
the deflection function for the + He system at 50.0 keV. At this energy, the elastic 
scattering is the only important process. Consequently, the classical elastic scattering 
theory can be used to invert the deflection function to yield the intermolecular potential. 
This potential is compared to the dynamic potential and to the Bom-Oppenheimer 
potential energy surface. However, before this comparison is made, we show the 
deflection function in Figure 4-18. 



68 




Figure 4-18: Deflection function for + He. ENDyne results at 50.0 eV with a pVDZ basis set. 

From the expression for the classical deflection function, equation 4,3, it is possible 
to devise a procedure that takes monotonic parts of the deflection function and through an 
inversion yields the potential function. This inversion procedure consists of the following 
steps [76, 63]: 



1. define some distance of closest approach s. 



2. evaluate numerically the integral 



I{s) = 7T 



1 



I 



^0 



0([s^ + 

\/ x'^ 



1 

/ 






y/y^S^ -I- 1/2 



(4.13) 



0 



3. compute r from 



r = 



(4.14) 



69 



4. determine a point on the potential energy curve (r, V'(r)) from the above expression 
for r and from the following expression for y(r) 



V(r) = E{1 - e"2^W). 



(4.15) 



5. determine values of V (r) at increasing separations r by gradually increasing the value 
of the lower limit s from which 0(6) is integrated. 

In order for this inversion procedure to be valid, the following conditions must be met. 
(i) Only the monotonic branches of the deflection function must used. In the case of a 
proton colliding with helium, the deflection function must be divided into three parts to be 
inverted, namely, {0, 6o), {6q, K), and {6r, oo}, and (ii) E > K(ro), that is, the potential 
can only be evaluated up to the classical turning point ro at the energy E, in addition 
(iii) only data that do not involve orbiting collisions may be inverted uniquely, that is, 

,,/ X rdV(r) 



When this procedure is applied to the deflection function. Figure 4-18, generated by 
ENDyne for + He at 50.0 eV, the potential curve in Figure 4-19 is obtained. 



70 




Figure 4-19: Potential energy curve obtained from the inversion of the deflection function 
generated by ENDyne for + He at 50.0 eV with a pVDZ basis set. 

During an evolution calculation the ENDyne program creates a file, called restart file, 
that contains the history of the system evolving in time. We then use another program, 
called EVOLVE, to extract relevant information about the evolution. At each instant in 
time during the evolution the position of each particle and the potential energy of the 
system are recorded. As a result, we can compute the dynamical potential, that is, the 
potential energy at each instant in time with its corresponding interatomic distance. This 
dynamical potential can be obtained from data before and after the collision, and also 
from several impact parameters. The data from both before and after the collision of 
proton -i- helium for impact parameters ranging from 0.01 to 1.0 a.u., yielded the same 
(up to six decimal places) potential curve. One of this curves is presented in Figure 4-20. 



71 




Figure 4-20: Potential energy curve obtained from the direct time evolution. ENDyne results for 
-I- He at 50.0 eV with a pVDZ basis set. 

We also used an ab initio CIS (configuration interaction with single excitations) to 
obtain the potential energy curves for the ground state and several excited states. The 
results for + He are presented in Figure 4-21. 



72 




Figure 4-21: Potential energy curves for + He obtained from a CIS calculation with a pVDZ 
basis set. 

Attempts to plot the ground state of a CIS calculation, together with the inverted 
deflection function and the dynamical potential show that they all lie on top of each other. 
This is a very reassuring result, since consistency between three different approach has 
been obtained. The reason for this agreement is due to the large separation between the 
potential curves as can be seen in Figure 4-21. This means that the dynamics will remain 
on only one surface and the non-Bom-Oppenheimer effects are negligible. In addition, 
unlike the H'*' -h H system, the energy to localize the electron around the helium atom 
in the incoming asymptotic channel is very small and the initial state basically is the 
ground state of the (H-t-He)"^ system. 



73 



Classical Differential Cross Section 

Recent improvements in data acquisition, analysis and detection techniques, error 
handling, and beam collimation have enabled experimentalist to explore regions of 
very small scattering angle [77, 78, 73]. The angular resolution achieved by these 
improvements is now good enough to show the oscillatory behavior of the ion-atom 
differential cross sections. Oscillations in the differential cross sections can be interpreted 
as due to interferences between different trajectories leading to the same scattering 
angle. Since our treatment of the nuclear motion is classical, we should not expect 
the appearance of these oscillations in the classical differential cross sections. The only 
structure appearing in the classical treatment is due to the oscillations in the probabilities 
for each process. 

In this subsection, we present the theoretical results obtained by the END-SD-FGWG 
method. The semiclassical corrections to our classical dynamics and comparisons with 
experimental cross sections are presented in the next subsection. 

We present the classical differential cross section for H'*’ + He at 500 eV in some 
detail, and just a summary of the results for 1500 and 5000 eV. Since the differential 
cross section depends upon the deflection function, we start by showing the three regions 
of the deflection function in Figure 4-22. The regions are, 

I) {0, 6o } where 6 q corresponds to the value of the impact parameter where the deflection 
function vanishes, the so-called glory angle, 0(6 q), 

II) ( 6o , 6r } where W defines the rainbow angle, that is, the extremum of the deflection 



function 



74 




and, 

in) {br,oo} the attractive region. 



(4.17) 




Figure 4-22: Deflection function for + He. ENDyne results at 500 eV with a pVDZ basis 
set. Regions: I) diamonds ’o’, II) triangles ’A’, and. III) open circles ’o’. 

As stated before, from the shape of the deflection function in Figure 4-22, we can 
see that more than one impact parameter leads to the same scattering angle 0. Thus, the 
expression for the classical differential cross section becomes 






t 



bi P{bi) 



sin 0 



dQ I 
dh l6=6i 



(4.18) 



75 



i.e., a sum of contributions from these different trajectories. The contributions of each one 
of the three regions for the elastic and for the charge transfer differential cross sections 
are shown in detail in Figures 4-23-4-24. 




Figure 4-23: Elastic differential cross section for H"" + He. ENDyne results at 500 eV with a 
pVDZ basis set. Regions: I) diamonds ’o’, II) triangles ’A’, and, III) open circles ’o’. 



76 




Figure 4-24: Charge transfer differential cross section for + He. ENDyne results at 500 eV 
with a pVDZ basis set. Regions: 1) diamonds ’o’, II) triangles ’A’, and, III) open circles ’o’. 

The elastic and charge transfer cross sections for + He at 500 eV, 1500 eV, and 
5000 eV are shown in Figures 4-25-4-27. 



77 




Figure 4-25: Elastic and charge transfer differential cross sections for + He. ENDyne results 
at 500 eV with a pVDZ basis set. 




Figure 4-26: Elastic and charge transfer differential cross sections for H* + He. ENDyne results 
at 1500 eV with a pVDZ basis set. 



78 




Figure 4-27: Elastic and charge transfer differential cross sections for + He. ENDyne results 
at 5000 eV with a pVDZ basis set. 

As we can see from Figures 4-25—4-27 the classical differential cross section becomes 
singular at the rainbow angle, and tends to infinity when the scattering angle goes to zero. 
We also should notice the difference in orders of magnitude between the elastic and the 
charge transfer processes. 

Semiclassical Elastic Differential Cross Section 



As mentioned before, the resolution of the experimental differential cross sections 
is high enough to allow us to observe quantum effects. Consequently, the results 
presented for the classical differential cross sections cannot directly be compared with 
the experimental data, except for the value of the rainbow angle. In order to remedy 
this problem we use semiclassical corrections [63, 75, 74, 79, 80] to our classical elastic 



differential cross sections. 



79 



We start by reviewing one of the old stablished techniques to solve the elastic 
scattering problem using time-independent methods . The elastic scattering by a spherical 
potential V{r) can be described by the wave function ^(f) which is an eigenfunction 



of the Schrodinger equation 



_ ^y(r) -f ^{f) = 0 



(4.19) 



with 



p = hk, k^ = 



(4.20) 



where p is the momentum. The elastic differential cross section can be expressed as 



follows 

^ = \Am 

where A{0) is called the scattering amplitude. 



(4.21) 



In order to obtain the scattering amplitude A{0) we expand the solution of the 



SchrOdinger equation for the scattering problem in terms of Legendre polynomials 



. OO 

^(^ = - Citpi{r)Pi{cos 6) 



(4.22) 



/=o 



where C/ are the amplitudes of the partial waves ipi{r), Pi{cos0) are the Legendre 
polynomials, and / is the angular momentum quantum number. This is the so called 
partial wave expansion, and it yields the following set of radial equations for V’/(r) 



- ?^V(r) - 
h H 



V’/(r) = 0 . 



(4.23) 



Using the normalization condition for ^'(f) and the orthogonality of the Legendre 
polynomials, we find that 

.. OO 

^ /=0 



(4.24) 



80 



where the phase shift is defined as 



6i = lim {Phase tpi{r) — Phase ‘4’i{r,V{r) = 0)} 



(4.25) 



Consequently, the crux of quantum mechanical scattering the calculation of the phase 
shift. Although the expression for the scattering amplitude, equation (4.24), is exact, the 
series converge extremely slowly. Several approximate techniques have been developed 
that give analytical expressions for the scattering amplitude. 

For instance, employing the WKB semiclassical approximation, the following ex- 
pression for the phase shift can be obtained [75], 



OC 




(4.26) 



where 




(4.27) 



and 




(4.28) 



Differentiating the WKB phase shift 6i with respect to / we obtain 



OO 




(4.29) 



To 

which when compared with the classical deflection function (a = tq) 



OO 




OO 



2J / drr-^FI~^^^{r) 



(4.30) 



81 



suggests that 




(4.31) 



This shows that the semiclassical phase shift is connected to the classical deflection 
function of the same angular momentum. Now that we have an analytical expression 
for the semiclassical phase shift we can use approximations of the Legendre polynomials 
Pi{cos0) to obtain expressions for the scattering amplitude. We analyze three cases, 
namely, a) angles not too small and not close to the rainbow angle Or, b) very small 
angles, and c) angles very near to Or. 

(a) {kb)~^ < 0 < TT {4 and 0 / Or- 

This range covers scattering at most angles, excluding very small angles and near the 
rainbow angle Or. In this range, the Legendre polynomials P/(cos 0) are approximated by 




2 



cos [(/ -t- 1/2)0 — 7r/4] + 0(/ ^) 



(4.32) 




and the WKB scattering amplitude can be simplified to 




(4.33) 



0 



with 




(4.34) 



Because of the small value of h these integrals may be evaluated by the method of 
stationary phase. After some algebraic manipulations, the semiclassical (sc) elastic 



82 



differential cross section is given by 



-SC 

= 



da 



2 cos(a, - oy) + E(‘'5')i 



(4.35) 



• • 
X<] 



where 



(4)i = - 



bi 



sin^ 



d© 

db 



(4.36) 



b=bi 



is the classical cross sections originating from each impact parameter that gives rise to 
the same scattering angle. Assuming that -tt < 0 < x, the phase difference can be 
expressed as 



Aq^ = a,- — Qj 



bi 



= k J db 0 ( 6 ) - ke{r]ibi - rjjbj) + ^ [( 77 , • - 27 /^;) - {T)j - 2 t)s.)] 
where rjs is the algebraic sign of the slope of the deflection function 



(4.37) 






(4.38) 



and 

, + 1 if 0(6,) > 0 

^ . (4.39) 

^ - 1 if 0(6,) < 0 

We find, therefore, that unless the deflection function 0 is a monotonic function of the 
impact parameter 6 and is confined to values between — x < 0 < x (so that for a 
given 6 there is but one value of 6), we do not get the classical result for the cross 
section, no matter how short the wavelength. Instead there are additional interference 
terms. However, for truly macroscopic scattering the beam of projectiles is not exactly 
monoenergertic. Thus, since the phase angles a,- contain the microscopic variable k, or 



83 



the wavelength, a spread in the energy, gives averaged phase angles yielding the purely 
classical result. 

(b) Very small angles: 9 < {kb)~^ ~ 0 and 9 ^ 9^. 

This range of scattering angle is called forward glory scattering. One of the classical 
trajectories which contributes to the forward glory corresponds to a large impact parameter 
(6i in Figure 4-18). The other is the glory trajectory at small b, corresponding the 
coalescence of 62 and 63 trajectories in Figure 4-18. The asymptotic expression for the 
Legendre polynomials Pi{cos9) at very small angles is 



0 

As a result, the scattering amplitude for very small angles can be expressed as 



P/(cos 9) ~ (cos 9yj() [(/ -|- 1 /2) sin 9\ ~ (cos 9)^Jq [(/ -|- 1 /2)0] (4.40) 



where the Bessel’s integral is 




(4.41) 




(4.42) 



We expand about the glory value, (0(6o) = 0, 



e{i) = e{io) + -^ , , (/-/o) + ... 

ai /=/o 



(4.43) 




and obtain the scattering amplitude 




db 6=60 




(4.44) 



84 



TTius, the elastic differential cross section for angles close to the forward glory angle 
is given by 





de 



dh 



b=bo 



[Jo(^^osin 0)]^. 



(4.45) 



(c) Near the rainbow angle: B k Or- 



The deflection function has an extremum at 0r, so we expand 0 about it 



0 = 0 r + 



1 SQ 



db^ 



b=br 






so that. 



c c 1 ^ / T T ^ 1 d^0 

h ^ h , + 56,(7 - 7,) + 



b=br 



(J-Jr)' 



The scattering amplitude close to the rainbow angle is given by 



A{B) ~ , 



1 






2irkb 
sin $ 



nl/2 



,«[a+»/fc6r(Sr-^)] 



Ai(.) 



where 



7 T 



Q = 26 b - kbrQr + TJT 

4 



and the Airy function is defined as 



00 



Ai( 2 ) = — J ducos{zu + 



0 



with the argument given by 



2 = 2^l^k^l^1)rn{0T - B) 



<Pe 



-1/3 



b=br 



Vt = sgn 



<i 62 



b=br 



(4.46) 



(4.47) 



(4.48) 



(4.49) 



(4.50) 



(4.51) 



85 



The value of the elastic differential cross section near the rainbow angle can then be 
written in the form 









25/Vibi/36r 

sin 6 



(Pe 

d9 



- 2/3 



|Ai(2))^ 



(4.52) 



The expression in equation (4.52) shows the complex structure of the elastic differential 
cross section in the region of Br when compared to the classical differential cross section. 
That is, the classical singularity at ^ has been smoothed out and replaced by an 
exponential damping 2tX B > Or and by an oscillatory feature at ^ This oscillating 
behavior arises from the Airy function for ^ < 0 and the maxima in the differential cross 
section, called rainbow maxima, correspond to the extrema in Ai( 2 ), that is, 2 = -1.0188, 
-3.2482, -4.8201, -6.1633, etc. Since, in general, 0(6 « 6r) < 0 and d‘^Q/db'^\i,=i^ > 0 



we have that 



2 = -{Br - B) 



1 d'^Q 



2P d9 



- 1/3 



h=hr 



= {B- Br)q-^l^ 



(4.53) 



1 <PQ 



9 = 



2it2 dip 



b=br 



Thus, the main peak occurs below Br, where 2 = -1.0188, i.e. 



Omax =Br- I.OISS?^/^ (4 54) 

Subsidiary supernumerary rainbows occur at lower angles (2 = -3.2482, -4.8201, 
-6.1633, etc). 

Experimental and Theoretical Elastic Differential Cross Sections 



The value of the rainbow angle, Br, can be determined experimentally as the position 
of the inflection in Od{B) beyond the rainbow maximum. For the scattering of a proton 



86 



with a helium atom the experimental determination of the rainbow angle is compared 
with the theoretical result in Table 4-5. The agreement between the ENDyne results and 
the experimental rainbow angles is excellent. 



Table 4-5: Experimental and theoretical rainbow angles for + He system. 



Energy 

(eV) 


Rainbow angle (degrees) 


Impact parameter 

(a.u.) 

ENDyne 


Experimental® 


ENDyne 


50.0 


— 


2.963 


1.826 


500 


0.32 


0.3015 


1.778 


1500 


0.11 


0.1013 


1.772 


5000 


0.03 


0.0302 


1.772 



a: Ref. [73]. 



We are working in the angular range of 0.01° — 1.0° and 0.04° < {kb)~^ < 0.29° 
for H'*’ + He at 500 eV. Consequently, we are in the regime of very small angles, 
^ ~ 0 and should use equation (4.45) for the differential cross section far from 

the rainbow angle. The semiclassical corrections to the classical elastic differential cross 
section obtained by the END-SD-FGWP method is shown in Figure 4-28. 



87 




Figure 4-28: Elastic differential cross section for + He. Energy: 5000 eV. Basis set: pVDZ. 
Solid line corresponds to the semiclassical corrections to the ENDyne results. Open circles are 
experimental data [73]. 

We have presented the results for 500 eV only since at this energy the elastic process 
is several orders of magnitude larger that the charge transfer. Thus, the semiclassical 
corrections to the classical elastic differential cross section should be good, as can be 
seen from the comparison between the theoretical and the experimental results. It remains 
to develop an appropriate formalism within END that yield these quantum interferences 
for inelastic processes. 

The H 2 Collision 

We have seen so far, that the END formalism based upon a single determinant and 
classical nuclei gives very good results for resonant and non-resonant ion-atom scattering. 
We now examine the performance of this method for ion-molecule collisions, in particular 

+ H 2 . In these type of collisions not only do the electrons play an important role 



88 



(transfer and excitation) but so do the nuclei, since vibrational and rotational excitations 
take place. There is a surprising lack (virtually none) of rigorous theoretical investigations 
on dynamics which occur in ion-molecule collisions in low- to intermediate-energy 
regime (50 eW < E < 50 keV). There are several reason for this lack of theoretical 
treatment, but mainly that, it is quite a complex problem to obtain reasonably accurate 
adiabatic potentials and eigenstates as functions of intemuclear distances and molecular 
orientations. In addition, for an ion-molecule system, the number of internal degrees 
of freedom that need a proper dynamical treatment increases dramatically. The END 
formalism in this context gains a lot of importance since it does not need adiabatic 
potentials and treats properly all degrees of freedom. In the present formulation, namely, 
END-SD-FGWP, we have not incorporated the ETF’s (electron translation factors), so 
it is limited to low-energies (E < I keV) applications. The END formalism is also 
capable of treating dynamics at very low-energies (E < 10 eV), which will be presented 
in Chapter 5. 

In this subsection we present the collision of a proton with a hydrogen molecule in 
its electronic and vibrational ground state at 500 eV. We examine some of the extreme 
molecular orientations, namely, (0°, 0°) and (90°, 0°). This should be representative of 
the effects of orientation of the target molecule on the charge-transfer mechanism, cross 
sections (charge transfer), and energy transfer mechanisms. 

Trajectory Analysis 

The nuclear motion of an ion-molecule collision is quite a bit more complicated (and 
more interesting) than the ion-atom scattering showed in the last two subsections. In an 



89 



ion-molecule collision one would expect that vibrational and rotational excitations occur. 
These excitations are shown graphically via the evolution of the nuclear coordinates 
in time. Figure 4-29 shows the head-on collision (0°, 0° orientation) for three impact 
parameters that lead to the same scattering angle (0.29°). We should clarify the notation 
for the molecular orientation (a, /3). The first angle, a, is the angle between the scattering 
direction and the molecule axis, and is the dihedral angle between the scattering plane 
and the molecular plane. Figure 4-29 represents the nuclear motions in the laboratory 
frame where the proton trajectories are the lines (almost straight lines) and the motion 
of the hydrogen atoms in the H 2 molecule are the oscillatory lines. A more detail 
representation of these motions, namely, the target atom coordinates, are plotted in 
Figure 4-30. 




Figure 4-29: Dynamical trajectories for + H 2 collision from ENDyne calculation for (0°, 0°) 
orientation at 500 eV with a pVDZ basis set. 



90 




Figure 4-30: Dynamical trajectories of the taiget H 2 . ENDyne results for (0°, 0°) orientation at 
500 eV with a pVDZ basis set. 

The nuclear motion in the laboratory frame for the (90°, 0°) orientation is plotted in 
Figure 4-31. These trajectories lead to the same scattering angle (0.29°) as in the case 
of (0°, 0°) orientation. As we can see, the trajectories for both molecular orientations 
are quite distinct, but a closer look shows the same qualitative behavior, i.e., at small 
impact parameters the target and projectile initially attract each other then the repulsive 
interaction overcomes this attraction and they move apart from each other. At large 
impact parameters they attract each other and move close together, but the target cannot 
keep up with the projectile velocity. In both cases the H 2 molecule starts vibrating and 
rotating after the collision. We should notice that even for the longest evolution times, 
namely, 1200 a.u. (= 30 fs) for (0.0°, 0.0°) and 800 a.u. (= 20 fs) for (90.0°, 0.0°), 
considered here, the rotational motion is almost absent. 



91 




X-Coordinate (a.u.) 



Figure 4-31: Dynamical trajectories for H"^ + H 2 collision from ENDyne calculation for (90°, 
0°) orientation at 500 eV with a pVDZ basis set. 




X'Coordinate (a.u.) 



Figure 4-32: Dynamical trajectories of the target H 2 . ENDyne results for (90°, 0°) orientation 
at 5(X) eV with a pVDZ basis set. 



92 



Vibrational Analysis 



The trajectory analysis seems to show that the H 2 molecular target vibrates in quite 
different vibrational states (frequency and amplitude). We emphasize again that the 
trajectories shown in the last subsection lead to the same scattering angle (0.29°). The 
angular dependence of the vibrational populations p„ of H 2 molecules after a collision 
with H''^ have been extensively studied by theoretical and experimental methods for low- 
(E < 30 eV) and high-energies (E > 10 keV). However, due to recent experimental 
developments vibrationally resolved measurements [81], differential in scattering angle 
6, have been made for the intermediate-energy range (100 eV < < 10 keV) and started 

challenging the theoreticians. Since we have a classical description of the nuclear motion, 
some approximate vibrational wavefunction would have to be determined to yield p„ as a 
function of 6. We have not fully analyzed this problem yet, so we present in this section 
the vibrational analysis of the classical motion of the nuclei. 

We find that the best way to obtain vibrational frequencies of the H 2 target is to use the 
Prony method [82] to analyze the interatomic distance of the H 2 molecule as function of 
time. Figure 4-33 shows the interatomic distances for all six impact parameters that lead 
to the same scattering angle (0.29°) for both (0°, 0°) and (90°, 0°) molecular orientations. 



93 




Figure 4-33: Interatomic distances of the target H 2 as function of time. ENDyne results for (0°, 
0°) and (90°, 0°) orientations at 5(X) eV with a pVDZ basis set. (1 a.u. = 0.024195 fs). 

The Prony method is an alternative to the DFT (discrete Fourier transform) for spectral 
analysis and is widely used in signal processing. It is a technique for modeling sampled 
data as linear combination of (complex) exponentials with damping coefficients e and 
angular frequencies uj 

V 

i(n) = (4.55) 

i=\ 



where the number of terms p is called the order of the method and the equally spaced 
steps of the data sample is n{6t). Details of the algorithm and its implementation can be 
found in reference [82]. Once the parameters of the exponential model are determined, 
the frequency spectrum is obtained directly, using the infinite domain analytic Fourier 



94 



transform 



N-l 

x{k) = N{6t) 

n=0 



(4.56) 



where N is the number of points in the data sample. In order to enhance the signal it 
is useful to use the relative spectral densities (RSD) as a function of the frequency u to 
obtain the spectrum. The RSD, expressed in decibel (dB), is defined as 

RSDHldbl = 101og.„(J^). (4.57) 



The vibration spectrum of H 2 for a specific trajectory (b = 1.25 a.u. and (a, = 

(0°, 0°)) is shown in Figure 4-34. The other trajectories present similar spectra. The 
largest order p of the method before linear dependency occurs was around 40 for all 
trajectories analyzed. 




Figure 4-34; Vibrational spectrum of the target H 2 . ENDyne results for (0°, 0°) orientation with 
impact parameter of 1.25 a.u. Eneigy: 500 eV. Basis set: pVDZ. 




95 



In Table 4-6 we present the vibrational frequencies for all trajectories shown in Figure 
4-33 (impact parameters leading to the same scattering angle (0.29°) for both (0°, 0°) and 
(90°, 0°) molecular orientations). The differences in the vibrational states are basically 
due to differences in the energy transfer and the electronic state of the H 2 target after 
the collision. 



Table 4-6: Vibrational frequencies (cm *) as function of the impact parameter (a.u.) and 
molecular orientation (a°, 13 °). 



b (a.u.) 


0 

0 


Vibrational Frequency 
(cm'^) 


1.25 


0,0 


4441 


1.59 


0,0 


4467 


3.42 


0,0 


4612 


1.70 


90,0 


4415 


2.20 


90,0 


4308 


3.50 


90,0 


4580 



Transfer Probability 



Since the vibrational ftequencies are different for a given scattering angle, it suggests 
that the electronic states of the H 2 target are also different. The transfer probability 
reflects these differences and is plotted in Figure 4-35. 



96 




Impact Parameter (a.u.) 



Figure 4-35: Probability for charge transfer in + H 2 collision. ENDyne results for (0°, 0°) 
and (90°, 0°) orientations. Eneigy: 500 eV. Basis set: pVDZ. 

It is interesting to note that even in these two extreme molecular orientations the 
transfer probability has similar qualitative behavior, however the quantitative differences 
are large enough to yield a variety of vibrational frequencies for the same scattering angle 

Deflection Function 

As we have seen before, the deflection function is an indicator of the appearance 
of the intermolecular potential. The adiabatic potential curves as a function of the 
- H 2 (center of mass) distance are highly dependent upon the molecular orientation for 
intermolecular distances smaller that 2.5 a.u. [83]. We should expect then some signif- 
icant differences between the deflection function for (0°, 0°) and (90°, 0°) orientations 
at small impact parameters. These deflection functions are shown in Figure 4-36. They 



97 



do confirm the findings of ab initio calculations for the dependence of the potential on 
the molecular orientation. 




Figure 4-36: Deflection functions for + H 2 collision. ENDyne results for (0°, 0°) and (90°, 
0°) orientations. Energy: 500 eV. Basis set: pVDZ. 

Differential Ooss Sections 

Since we have the probabilities and the deflection function we can compute the 
classical differential cross section for these two molecular orientation. The calculation of 
the differential cross sections is also motivated by the experimental measurements [84] 
of the transfer and elastic processes in + H 2 collision. 

Figures 4-37—4-38 show the elastic and charge transfer differential cross sections 
calculated by ENDyne for the two molecular orientations considered here and compared 
with the experimental data [84]. 



98 




Figure 4-37: Elastic differential cross section for + H 2 collision. ENDyne results for (0°, 
0°) and (90°, 0°) orientations. Experimental data from reference [84]. Energy: 500 eV. Basis 
set: pVDZ. 




Figure 4-38: Charge transfer differential cross section for + H 2 collision. ENDyne results 
for (0°, 0°) and (90°, 0°) orientations. Experimental data from reference [84]. Energy: 500 
eV. Basis set: pVDZ. 



99 



The calculated classical elastic differential cross sections for both molecular orien- 
tations agree quite well with the experimental results. From the inflection in the exper- 
imental data for the elastic cross section we estimate that the rainbow angle should lie 
between 4.0° and 4.7°, which agrees quite well with the ’averaged’ theoretical rainbow 
angle from the molecular orientations considered here. The agreement with the experi- 
mental data for the charge transfer cross section is not as good as for the elastic one. The 
reason perhaps is that we are comparing classical cross sections with the experimental 
results. The latter, by nature, includes the quantum effects due to interference between 
different trajectories leading to the same scattering angle. In the case of a molecular 
target, these effects should be even greater than for atomic target, since we have also 
interference between trajectories from different molecular orientations. 

Summary and Conclusions 



Both quantum and classical dynamical quantities calculated by the END-SD-FGWP 
method is very satisfactory. Some of the quantum dynamical quantities like transfer 
probability and interparticle interactions are compared with experimental results and it 
shows an excellent agreement. So do some classical quantities like rainbow angles. 
However, some work has to be done to develop appropriate techniques and formalisms 
to correct classical results like vibrational populations and quantum effects in differential 
cross sections. 

The systems treated here + H, He, and H 2 present a wide variety of situations. 
For instance, we have resonant (H'*' + H), near-resonant (H"^ - 1 - H 2 ), and non-resonant (H'*’ 



100 

+ He) situations, and the END-SD-FGWP method represented them quite well. Also, 
we have shown that the molecular orientation plays an important role in charge transfer 
processes as do the quantum effects due to contributions from different trajectories. 



CHAPTER 5 

INTRAMOLECULAR CHARGE TRANSFER DYNAMICS 



The dynamics of inter- and intramolecular charge transfer has been a very important 
subject in chemical-physics sciences [85-89]. The main reason is that charge transfer is 
the essence of some most fundamental biological and biochemical processes. Also, charge 
transfer is important in many fields such as atomic physics, plasma physics, astrophysics, 
semiconductor physics, organic and inorganic chemistry, and many others. 

In this chapter we study the following charge transfer systems Li — H-Li'*' ^ Li"^- 
H — Li and Li-CN-Li"^ ^ Li'^-CN-Li. These molecular systems represent a good prototype 
for theoretical studies of charge transfer dynamics. They are small enough to be treated at 
some high level ab initio time-dependent and time-independent methods. In addition, the 
Li-CN-Li system can be seen as exhibiting charge transfer between two mixed valence 
metal atoms mediated by a bridge. 

Electron Transfer Formalisms 

Due to its conceptual simplicity the formalism for electron transfer proposed by 
Marcus [90-92, 89] in the late 50’s has drawn a lot of attention from both theoreticians 
and experimentalists. This theory has been extensively reviewed, revised, and extended. 
However, two basic assumptions are still being employed, namely, a) that there is a 
reaction coordinate that takes the reactants to the products via nuclear motion, and b) 
there is a coupling (// 12 ) between two electronic states (donor and acceptor). A typical 
electron transfer reaction coordinate is shown in Figure 5-39. 



101 



102 




Figure 5-39: Diabatic and adiabatic potential curves for normal electron transfer. 

Non-dynamical type approaches for electron transfer assume that the transfer rate k 
is usually given by an expression based on the Fermi golden rule [92, 89, 93] 

2ir 

»: = y|/ri2p(FC). (5.1) 

The quantity (FC) is the Franck-Condon factor and is related to the vibrational spectrum 
of the donor-acceptor system and its surroundings, if any. Analytical expressions for (FC) 
derived from classical and semiclassical theories exist and are discussed in references [92, 
89]. The matrix element H \2 represents the electronic coupling of the donor and acceptor 
(or reactant and product). This is the quantity that has been challenging theoreticians and 
experimentalists. There are several approaches to compute the electron transfer matrix 
element. Usually, for large systems such as proteins, the transfer dynamics is treated as 
a one-electron problem [87, 93]. However, for small molecular systems many-electrons 



103 



ab initio and semiempirical theories have been used [89]. In this case, the most common 
approach to calculate H \2 is to compute the diabatic states which, in general, are non- 
orthogonal and use them as zeroth order states in a perturbational approach [94, 89] or 
use them in a two-dimensional non-orthogonal Cl (configuration interaction) problem 
[95, 96]. More specifically, the broken space symmetry wave functions, |1) and |2), are 
used in the following expression for the coupling element [97, 94, 96, 89] 






1 



1 Si2 



{\\H\2)-Sn 



(l|/f|l)-H(2|i/l2) 



(5.2) 



5i2 = (1|2). 



Since the nuclear motion is responsible for electron transfer [97, 89], some attempts 
have been made to treat the motion of the nuclei explicitly. Most of these dynamical 
approaches are also two-state models with a coupling H\ 2 . One of these approaches uses 
an ab initio method to calculate the coupling H 12 at each nuclear configuration as the 
system is evolving in time [98, 99]. Some simpler approaches use a model potential 
function for the electronic state and the nuclear motion is treated explicitly by the time- 
dependent variational principle [100] or by the semiclassical approach of Nikitin [101]. 
All these approaches for electron transfer need to compute the coupling matrix element. 
In order to obtain H 12 it is necessary to know first what is the reaction coordinate, and 
that might be very difficult to determine for large systems. 

We use the ENDyne program based on the END-SD-FGWP formalism to study 
intramolecular electron transfer. The main advantage is that we do not need to compute 



104 



the coupling matrix element, and to initialize the electron transfer we just distort the 
molecule and let the system evolve in time. 

Structure, Energetics, and Electron Density of LiCNLi 



Before presenting the results for LiCNLi we should specify that all results presented 
in this subsection were obtained using the 3-21G basis set [102]. In addition, we 
perform time-independent ab initio self-consistent field (SCF) calculations to determine 
potential surfaces and electronic densities of the LiCNLi and LiHLi molecules. We 
do not describe the details of the Hartree-Fock method and the methods that introduce 
electronic correlation, since we use standard approaches widely tested and reviewed in 
the literature [103-105]. 

The potential energy surface of the LiCNLi molecule has been recently studied where 
the basis set and electronic correlation effects were studied in some detail [106]. We 
augmented those results by adding data from a UHF/3-21G calculation. The linear 
structure of LiCNLi has two minima at all levels of theory explored. Table 5-7 presents 
the results for these two linear structures at several levels of theory. As we can see 
the general geometrical structure of linear LiCNLi is quite independent of basis set and 
electronic correlation corrections. 



105 



Table 5-7: Minima for the linear structure of LiCNLi at various levels of theory. Bond distance 
in pm. 



Theory Level 




Structure P 




Structure IF 


1 


LiC 


CN 


NLi 


LiC 


CN 


NLi 


UHF/3-21G 


198.0 


115.0 


192.6 


214.1 


115.8 


178.8 


UHF/6-31G* 


198.3 


114.5 


198.8 


216.6 


115.3 


182.1 


UHF/6-31+G* 


196.5 


114.6 


197.9 


215.9 


115.4 


180.7 


UMP2/6-31G* 


196.0 


118.4 


200.1 


213.1 


118.8 


183.6 


CASS/3-21G 


197.0 


117.4 


194.3 


212.0 


117.7 


180.5 



a: All results from Ref. [106] except UHF/3-21G. 



These two linear structures are local minima in the potential energy surface. The 
proposed global minimum has a C 2 v structure at UHF/3-21G level with the lithium 
atoms bonded to the nitrogen. Another interesting fact about the potential energy surface 
of this system is that the transition state between the linear structures (I and II) has Cg 
symmetry. The structure of this transition state and global minimum at the UHF/3-21G 
level is presented in Figure 5-40. 




Transition Slate I-D 




Figure 5-40: Transition state for structures I and II and global minimum structures. Bond 
distances in pm and angles in degrees. 



The energetics of the extrema of the potential energy surface of LiCNLi is indicated 



106 



in Table 5-8. As we can see, the qualitative and quantitative features of the potential 
energy surface of LiCNLi are quite basis set independent, and only the relative stability 
of the linear structures depends upon the theoretical level of electronic correlation. 



Table 5-8; Relative energies of minima I and II and of transition state I-II (Cs). Eneigies in 
kcal/mol. 



Theory Level® 


Structure I 


Structure II 


Transition 

State 


Global 

Minimum 


UHF/3-21G 


8.68 


4.94 


9.11 


0.0 


UHF/6-31+G* 


9.78 


7.00 


11.58 


0.0” 


UMP2/6-31G* 


7.75 


9.00 


9.15 


4D 

O 

• 

o 



a: AU results from Ref. [106] except UHF/3-21G, b: The global minimum has a Cj structure. 

We think that the LiCNLi molecule is a good prototype system for charge transfer 
dynamics. We present Mulliken population analysis, dipole moments, and isodensity 
surfaces for all extrema in the potential energy surface of LiCNLi. Table 5-9 contains 
charges and dipole moments for these structures. As we see from Table 5-9 almost a 
complete electron transfer takes place when going from structure I to II and vice versa. 
Since the atomic charge is not an observable, we plot in Figures 5-41-5-44 the electron 
isodensity surfaces of these structures for both alpha and beta spins, and confirm what 
has been stated using ordinary Mulliken analysis. We should mention that the assignment 
of alpha or beta electrons is arbitrary, and we choose to have ten alpha electrons and 9 
beta electrons. As a result, assuming that the spin contamination is small, the unpaired 
electron is alpha and this explains why the alpha electron density is asymmetric. 



107 



Table 5-9: MuUiken population (a.u.) and dipx)le moments (D) of structures I and II, transition 
state I-II and global minimum at UHF/3-21G level. 





Structure I 


Structure II 


Transition 

State 


Global 

Minimum 


Li(C) 


0.704 


-0.168 


0.017 


0.283 


C 


0.061 


0.213 


0.138 


0.256 


N 


-0.599 


-0.761 


-0.674 


-0.821 


Li(N) 


-0.167 


0.716 


0.518 


0.283 


Dipole (D) 


-16.74 


15.68 


-6.16 


-0.54 



alpha density 



beta density 



Figure 5-41: Alpha and beta isodensity for structure I at UHF/3-21G level. Isodensity 

is 0.003 a.u. (1 a.u. = 6.748 e/A3|. Structure: Li-C = 197.98 pm (= 1.980 A = 3.741 
a.u.), C-N = 115.02 pm (= 1.150 A = 2.174 a.u.), Li-N = 192.59 pm (= 1.926 A = 
3.639 a.u.). SCF energy: -106.63313290 a.u. = -2901.6368 eV 





alpha density 



beta density 



Figure 5-42: Alpha and beta isodensity for structure II at UHF/3-21G level. Isodensity 

is 0.003 a.u. (1 a.u. = 6.748 e/A3|. Structure: Li-C = 214.10 pm (= 2.141 A = 4.046 
a.u.), C-N = 115.77 pm (= 1.158 A = 2.188 a.u.), Li-N = 178.79 pm (= 1.788 A = 
3.379 a.u.). SCF energy: -106.63909258 a.u. = -2901.7989 eV 






ti cn 



109 





alpha density 



beta density 



Figure 5-43: Alpha and beta isodensity of the transition state I-II structure (Cs) at UHF/3-21G 

level. Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). Structure: Li-C = 208.75 pm (= 2.088 A 
3.945 a.u.), C-N = 115.83 pm (= 1.158 A = 2.189 a.u.), Li-N = 181.89 pm (= 1.819 A = 
.437 a.u.), Li-C-N = 148.49°, C-N-Li = 138.61°. SCF energy: -106.63244750 a.u. = 
-2901.6180eV 





alpha density 



beta density 



Figure 5-44: Alpha and beta isodensity of the global minimum structure (C2v) at UHF/3-21G 

level. Isodensity is 0.003 a.u. (1 a.u. = 6.748 e/A^). Structure: C-N = 117.46 pm (= 1.175 A 
= 2.220 a.u.), Li-N = 189.22 pm (= 1.892 A = 3.576 a.u.), C-N-Li = 137.12°. SCF energy: 
-106.64697039 a.u. = -2902.0134 eV 






no 



The choice of an isodensity value of 0.003 a.u. (1 a.u. = I/oq = 6.748 e/A^) is 
based on the fact that the isodensity in the range of 0.002-0.003 a.u. yields molecular 
dimensions that agree with kinetic theory data, so the expected shape of the molecule 
can be shown [107]. Figures 5-41 and 5-42 corroborate the finding of the Mulliken 
population analysis and the dipole moment calculations for structures I and II. 

We are unable to find any transition state that is linear. Consequently, the reaction 
coordinate for the electron transfer is non-trivial and application of a two-state type 
model is not obvious. As a result, we decided not to perform any dynamical calculation 
and instead to explore the intramolecular electron transfer in the LiHLi molecule. The 
behavior of the LiHLi molecular wave function and potential surface is common among 
the electron transfer systems and dynamical results from the END-SD-FGWP formalism 
can be compared with other theoretical approaches. 

Structure, Energetics, and Electronic Population of LiHLi 

Before presenting the results for LiHLi we should specify that all results presented 
in this subsection were obtained using the 3-21G basis set for H [102] and 3-214G for 
Li [108]. In addition, we perform time-independent ab initio self-consistent field (SCF) 
calculations to determine potential surfaces and electronic densities of the LiHLi and 
LiCNLi molecules. 

The linear structure of the open shell LiHLi molecule at the SCFAJHF level has 
a broken symmetry wave function which is strongly (charge) localized. This can be 
seen in Figure 5-45, where we plot the difference of Mulliken charges on the lithium 



Ill 



atoms versus the relative Li-H bond distances, and the potential energy as function of 
the reaction coordinate in Figure 5-46. 




Figure 5-45: Mulliken charge differences at SCFAJHF level in the Li(l)-H-Li(2) molecule as 
function of relative bond distances (rj - T 2 ). 



112 




Figure 5-46: Potential energy curves of the Li(l)-H-Li(2) molecule at SCF/UHF level. 

As we can see from Figures 5-45-5-46, the broken symmetry electronic wave function 
(lower surface) exhibits an abrupt change at the symmetric structure (ri - r 2 = 0, where ri 
= distance Li(l)-H and r 2 = distance H-Li(2)), This leads to an unphysical discontinuous 
behavior of the charge density. The wave function describing the upper surface does not 
show this behavior perhaps because it has less spin contamination. We should mention 
that the two asymmetric linear structure minima are in fact saddle points, since they have 
one imaginary vibrational frequency leading to the bend structure. Nevertheless, the linear 
structures are a good candidate for charge transfer studies since the system is small enough 
to allow us to perform some computational experiments within the ENDyne program. 

Dynamics of Electron Transfer in LiHLi 



We present the dynamical results for the intramolecular electron transfer in LiHLi 



113 



obtained using the ENDyne program within the END-SD-FGWP formalism. 

The procedure utilized here is the following, 

(1) make a distortion of the molecular geometry (contract and/or stretch bonds) relative 
to the linear minima structures; 

(2) for this non-equilibrium geometry perform a SCF/UHF calculation and use the 
converged molecular orbitals as the initial z parameters; 

(3) let the system evolve in time and analyze the molecular properties as function of time. 

We have performed time-dependent calculations for four different initial conditions 
(geometry and electronic wave function), namely, asymmetric stretch and contraction of 
the Li-H bonds by 5.6%, 10%, and 15%, and an almost symmetric structure (Iri - T 2 \ 
= 0.004 a.u.). Table 5-10 presents the structure and energies of these initial geometries 
for evolution. 



Table 5-10: MuUiken population, structure, and relative energy of the LiHLi molecule at 
SCF/UHF level. 



Structure 


MuUiken population 


Bond distances (a.u.) 


Relative 

energy 

(kcal/mol) 




Li(l) 


Li(2) 


Li(l)-H 


Li(2)-H 


Minimum 


2.387 


3.243 


3.0904 


3.4559 


0.0 


5.6% 


2.478 


3.195 


2.9170 


3.6491 


0.97 


10% 


2.498 


3.186 


2.7814 


3.8015 


3.2 


15% 


2.522 


3.178 


2.6110 


3.9743 


8.03 


SYM 


2.436 


3.219 


3.2549 


3.2586 


0.87 



The results of the time evolution of these structures (Table 5-10) are presented in 



114 



Figures 5-47-5-50. The changes of the Li-H bond distances as functions of time are 
plotted in Figure 5-47. 




Figure 5-47: Bond distances, ri = distance Li(l)-H and tz = distance H-Li(2) in a.u. as a 
function of time. ENDyne calculation with a H/3-21G and Li/3-21+G basis set. 

Figure 5-48 has the results for the Mulliken population of the lithium atoms during the 
evolution, and the differences in the Mulliken atomic charges are plotted in Figure 5-49. 



115 




Figure 5-48: Atomic Mulliken population of Li(l) and Li(2) as a function of time. ENDyne 
calculation with a H/3-21G and Li/3-21+G basis set. 



116 




Figure 5-49: Atomic Mulliken charge differences q[Li(l)]-q[Li(2)] as a function of time. 
ENDyne calculation with a H/3-21G and Li/3-2 1+G basis set. 

Comparing the atomic charges as functions of time in Figure 5-49 with the charges in 
terms of the reaction coordinate in Figure 5-45 we see that the dynamical treatment seems 
to better describe the charge transfer since it does not exhibits unphysical discontinuity 
like the time-independent calculation. 



117 




Figure 5-50: Alpha and beta spin atomic Mulliken populations of Li(l) and Li(2) in the 15% 
stnicture as a function of time. ENDyne calculation with a H/3-21G and Li/3-2 1+G basis set. 

Figure 5-50 represents a more detailed Mulliken population for the 15% structure. 
As we can see the beta spin populations remain quite constant in time and all the electron 
transfer occur between the alpha spin electronic densities. The reason for this is that, as 
we said before, the assignment of alpha or beta spin for the electrons is arbitrary, and as 
in the case of LiCNLi, we have assigned alpha spin to the unpaired electron. 

Summary and Conclusion 



In conclusion of this chapter we again point out that the application of the END-SD- 
FGWP method to electron transfer does not depend upon any coupling matrix element 
neither is an a priori knowledge of the reaction coordinate necessary. In addition the 
ENDyne results for intramolecular charge transfer in the LiHLi molecules are smooth in 



118 

both time and nuclear displacement, unlike the SCF/UHF results that show an unphysical 
discontinuity at the symmetric structure. 



CHAPTER 6 
CONCLUSIONS 



The exploration of the capabilities of the END-SD-FGWP method show that it is 
successful in providing an appropriate description of electron nuclear dynamics. The 
main advantages of the END formalism are that it (i) does not need the knowledge of 
potential energy surfaces, (ii) treats appropriately the coupling between electrons and 
nuclei, and (iii) does not require a priori knowledge of the reaction coordinate neither 
the coupling matrix element for calculating electron transfer. 

The approximations employed, namely a single determinantal wave function for the 
electrons and a classical description of the nuclei seem to be excellent starting points for 
generating qualitative and in some cases even quantitative results in molecular dynamics. 
For instance, in the case of the intramolecular electron transfer in LiHLi, the Mulliken 
charges generated by the ab initio SCFAJHF method show an unphysical discontinuous 
behavior at the linear symmetric structure. The dynamical calculations, however, do 
not exhibit this behavior. In fact, the Mulliken charges, calculated by the ENDyne 
program based upon the END-SG-FGWP formalism, change with a continuous behavior. 
Although, these are successful results, there is still room for further developments, from 
both formal and computational. 

Inclusion of electron correlation with an multireference wave function [2] would bring 
the END formalism to a level where, predictive, quantitative results could be obtained for 
most modest size molecular system. As we have seen from the scattering results of the 



119 



120 



+ He and + H 2 systems the quantum interferences between different trajectories 
are important for an appropriate description of the differential cross sections. We present 
semiclassical corrections to this problem, for the elastic differential cross section, which 
seem to work well. However, for inelastic scattering the semiclassical corrections cannot 
be straightforwardly employed. Thus, an appropriate approach within the END-SD- 
FGWP formalism needs to be developed in order to take into account these quantum 
interference effects. The proper way of treating this problem would be the extension 
to include a quantum description of the nuclei. In other words, if we avoid the FGWP 
(classical) approximation we could solve the problem of quantum interferences in the 
differential cross section calculations. This quantum treatment of the nuclei would also 
provide the appropriate approach for processes such as nuclear tunneling, multi-channel 
reactions, and zero point energy effects. 

The ab initio implementation of the END-SD-FGWP method requires the evaluation 
of all integrals and integral derivatives for each step in the differential equation solver. 
This makes the ab initio implementation computationally demanding, and limits the 
applications to small and medium sized molecules. We have presented a simplification 
of the END-SD-FGWP method that uses the NDDO approximation with effective cores. 
An appropriate parametrization of the NDDO, such as provided by the AMI and PM3 
methods has proved useful and accurate in describing stationary molecular properties. It 
is our expectation that the same quality of description will also be adequate in the case of 
time-dependent methods. From a computational point of view the savings will be twofold: 
(i) a decrease by several orders of magnitude of the number of molecular integrals, (ii) 



121 

the elimination of high frequency motions (core electrons) allowing the integrator to take 
larger time steps, and (iii) a decrease of the number of dynamical equations. 



REFERENCES 



1. P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930). 

2. E. Deumens, Y. Ohm, and B. Weiner, J. Math. Phys. 32, 1166 (1991). 

3. A. D. Hammerich, R. Kosloff, and M. A. Ratner, Chem. Phys. Lett. 171, 97 (1990). 

4. J. A. Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw- 
Hill, New York, 1970. 

5. M. J. Cohen, N. C. Handy, R. Hernandez, and W. H. Miller, Chem. Phys. Lett. 192, 
407 (1992). 

6. M. Karplus, R. Porter, and R. D. Sharma, J. Chem. Phys. 49, 3259 (1965). 

7. J. D. Doll, Chem. Phys. 3, 257 (1974). 

8. L. M. Raff and D. L. Thompson, The classical trajectory approach to reactive 
scattering, in The Theory of Chemical Reaction Dynamics, edited by M. Baer, pages 
1-121, CRC Press, Boca Raton, FL., 1985. 

9. T. F. George, K. T. Lee, W. C. Murphy, M. Hutchinson, and H. W. Lee, Theory of 
reactions at a solid surface, in The Theory of Chemical Reaction Dynamics, edited by 
M. Baer, pages 139-169, CRC Press, Boca Raton, FL., 1985. 

10. J. C. Tully, Adv. Chem. Phys. 42, 63 (1980). 

11. B. J. Kuntz, Semiempirical potential energy surfaces, in The Theory of Chemical 
Reaction Dynamics, edited by M. Baer, pages 71-90, CRC Press, Boca Raton, FL., 
1985. 

12. E. J. Heller, J. Chem. Phys. 62, 1544 (1975). 

13. E. J. Heller, Chem. Phys. Lett 34, 321 (1975). 



122 



123 



14. E. J. Heller, J. Chem. Phys. 65, 4979 (1976). 

15. E. J. Heller, Acc. Chem. Res. 14, 386 (1981). 

16. D. A. Micha, J. Chem. Phys. 78, 7138 (1983). 

17. C. D. Stodden and D. A. Micha, Int. J. Quantum Chem.. S21, 239 (1987). 

18. S.-Y. Lee and E. J. Heller, J. Chem. Phys. 76, 3035 (1982). 

19. R. H. Bisseling et al., J. Chem. Phys. 87, 2760 (1987). 

20. R. Kosloff, J. Phys. Chem. 92, 2087 (1988). 

21. C. LeForestier et al., J. Comput. Physics 94, 59 (1991). 

22. K. C. Kulander, Comput. Phys. Commun. 63, 1 (1991). 

23. M. Kimura and W. R. Thorson, Phys. Rev. A 24, 1780 (1981). 

24. W. Fritsch and C. D. Lin, Phys. Rep. 202, 1 (1991). 

25. M. Kimura and N. F. Lane, The low-energy, heavy-particle collisions - a close- 
coupling treatment, in Advances in Atomic, Molecular and Optical Physics, edited by 
D. Bates and B. Bederson, page 79, Academic Press, New York, 1990. 

26. M. Kimura, Phys. Rev. A 44, R5339 (1991). 

27. M. Karplus and J. A. McCammon, Annu. Rev. Biochem. 53, 263 (1983). 

28. P. A. Kollman and K. M. Merz, Acc. Chem. Res. 23, 246 (1990). 

29. W. A. Eaton and A. Szabo, Chem. Phys. 158, 191 (1991). 

30. M. Parrinello, First-principles molecular dynamics, in Modern Techniques in 
Computational Chemistry: MOTECC-90, edited by E. dementi, pages 731-743, 
ESCOM, Leiden, The Netherlands, 1990. 

31. P. A. Bash, M. Field, and M. Karplus, J. Amer. Chem. Soc. 109, 8092 (1987). 



124 



32. K. R. S. Devi and S. E. Koonin, Phys. Rev. Lett. 47, 27 (1981). 

33. K. C. Kulander, K. R. S. Devi, and S. E. Koonin, Phys. Rev. A 25, 2968 (1982). 

34. K. Runge, D. A. Micha, and E. Q. Feng, Int. J. Quantum Chem. S24, 781 (1990). 

35. D. H. Tiszauer and K. C. Kulander, Comput. Phys. Commun. 63, 351 (1991). 

36. M. J. Field, J. Chem. Phys. 96, 4583 (1992). 

37. B. Hartke and E. A. Carter, Chem. Phys. Lett. 189, 358 (1992). 

38. E. Deumens, A. Diz, H. Taylor, and Y. Ohm, J. Chem. Phys. 96, 6820 (1992). 

39. A. Diz, Electron Nuclear Dynamics: A Theoretical Treatment Using Coherent States 
and the Time-Dependent Variational Principle., PhD thesis. University of Florida, 
Gainesville, FL, 1992. 

40. J. R. Klauder and B.-S. Skagerstam, Coherent States, Applications in Physics and 
Mathematical Physics, World Scientific, Singapore, 1985. 

41. A. D. McLachlan and M. A. Ball, Rev. Mod. Phys. 36, 844 (1964). 

42. P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle 
in Quantum Mechanics, Springer, New York, 1981. 

43. J. Broeckhove, L. Lathouwers, E. Kesteloot, and P. Van Leuven, Chem. Phys. Lett. 
149, 547 (1988). 

44. M. C. Zemer, Mol. Phys. 23, 963 (1972). 

45. J. A. Pople, D. P. Santry, and G. A. Segal, J. Chem. Phys. S43, 129 (1965). 

46. R. G. Parr, J. Chem. Phys. 20, 239 (1952). 

47. K. R. Roby, Chem. Phys. Utt. 11, 6 (1971). 

48. K. R. Roby, Chem. Phys. Lett. 12, 579 (1972). 



125 



49. K. F. Freed, J. Chem. Phys. 60, 1765 (1974). 

50. R. Iffert and K. Jug, Theor. Chim. Acta 72, 373 (1987). 

51. K. Riidenberg, J. Chem. Phys. 19, 1433 (1951). 

52. P. O. Lowdin, J. Chem. Phys. 21, 374 (1953). 

53. M. S. J. Dewar, E. G. Zoebisch, E. F. Healy, and J. J. P. Stewart, J. Am. Chem. 
Soc. 107, 3902 (1985). 

54. W. Thiel, Tetrahedron 44, 7393 (1988). 

55. J. J. P. Stewart, Semiempirical molecular orbital methods, in Reviews in Computa- 
tional Chemistry, edited by K. B. Lipkowitz and D. B. Boyd, pages 45-81, VCH 
Publishers, New York, 1990. 

56. M. S. J. Dewar and W. Thiel, J. Am. Chem. Soc. 99, 4899 (1977). 

57. P. Hoggan and D. Rinaldi, Theor. Chim. Acta 72, 47 (1987). 

58. T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 

59. W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 56, 2257 (1972). 

60. T. H. Dunning, Jr., J. Chem. Phys. 53, 2823 (1970). 

61. R. C. Weast, Handboodk of Chemistry and Physics, CRC Press, Boca Raton, FL, 
67 edition, 1986. 

62. J. P. Davis and W. R. Thorson, Can. J. Phys. 56, 996 (1978). 

63. G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham, Intermolecular Forces. 
Their Origin and Determination, Clarendon Press, Oxford, 1981. 

64. J. N. Murrell and S. D. Bosanac, Chem. Soc. Rev. 95, 21 (1992). 

65. J. H. Newman et al., Phys. Rev. A 25, 2976 (1982). 



126 



66. M. W. Gealy and B. Van Zyl, Phys. Rev. A 36, 3091 (1987). 

67. J. D. Jackson, Classical Electrodynamics, John Wiley, New York, 2nd edition, 1975. 

68. J. C. Houver, J. Fayeton, and M. Barat, J. Phys. B: At. Mol. Phys. 7, 1358 (1974). 

69. H. F. Helbig and E. Everhart, Phys. Rev. 140, 715 (1965). 

70. C. Gaussorgues, C. L. Sech, F. Masnou-Seeuws, R. McCarroll, and A. Riera, J. 
Phys. B: At. Mol. Phys. 8, 253 (1975). 

71. M. Kimura, Phys. Rev. A 31, 2158 (1985). 

72. R. L. Fitzwilson and E. W. Thomas, Phys. Rev. A 6, 1054 (1972). 

73. L. K. Johnson et al., Phys. Rev. A 40, 3626 (1989). 

74. R. M. C. McDowell and J. P. Coleman, Introduction to the Theory of I on- Atom 
Collisions, North-Holland, Amsterdam, 1970. 

75. R. G. Newton, Scattering Theory of Waves and Particles, Springer- Verlag, New 
York, 1982. 

76. U. Buck, J. Chem. Phys. 54, 1923 (1971). 

77. D. E. Nitz, G. S. Gao, L. K. Johnson, K. A. Smith, and R. F. Stebbings, Phys. Rev. 
A 35, 4541 (1987). 

78. G. S. Gao, L. K. Johnson, J. H. Newman, K. A. Smith, and R. F. Stebbings, Phys. 
Rev. A 38, 2789 (1988). 

79. K. W. Ford and J. A. Wheeler, Ann. Phys. 7, 259 (1959). 

80. M. V. Berry, Proc. Phys. Soc. (London) 89, 479 (1966). 

81. D. Dhuicq and C. Benoit, J. Phys. B: At. Mol. Opt. Phys. 24, 3588 (1991). 

82. S. L. Marple, Jr., Digital Spectral Analysis, Prentice-Hall, Englewood Cliffs, New 
Jersey, 1987. 



127 



83. M. Kimura, Phys. Rev. A 32, 802 (1985). 

84. R. S. Gao et al., Phys. Rev. A 44, 5599 (1991). 

85. B. Chance et al.. Tunneling in Biological Systems, Academic Press, New York, 1979. 

86. D. C. De Vault, Quantum-Mechanical Tunneling in Biological Systems, Cambridge 
University, New York, 2nd edition, 1984. 

87. D. N. Beratan, J. N. Onuchic, J. N. Betts, B. E. Bowler, and H. B. Gray, J. Am. 
Chem. Soc. 112, 7915 (1990). 

88. J. J. Hopfield, J. N. Onuchic, and D. N. Beratan, Science 241, 817 (1988). 

89. M. D. Newton, Chem. Rev. 91, 767 (1991). 

90. R. A. Marcus, J. Chem. Phys. 24, 966 (1956). 

91. R. A. Marcus, J. Chem. Phys. 46, 679 (1965). 

92. R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 (1985). 

93. J. W. Evenson and M. Karplus, J. Chem. Phys. 96, 5272 (1992). 

94. K. Ohta, G. L. Closs, K. Morokuma, and N. J. Green, J. Am. Chem. Soc. 108, 1319 
(1986). 

95. C. F. Jackels and E. R. Davidson, J. Chem. Phys. 64, 2908 (1976), Non-orthogonal 
Cl. 

96. A. Broo and S. Larsson, Chem. Phys. 148, 103 (1990). 

97. M. D. Newton, Int. J. Quantum Chem. S14, 363 (1980). 

98. K. V. Mikkelsen, E. Dalgaard, and P. Swanstrom, J. Phys. Chem. 91, 3081 (1987). 

99. K. V. Mikkelsen and M. A. Ratner, J. Phys. Chem. 93, 1759 (1989). 

100. E. Deumens, Y. Ohm, and L. Lathouwers, Int. J. Quant. Chem. S21, 321 (1987). 



128 



101. S. Larsson, Theor. Chim. Acta 60, 111 (1981). 

102. J. S. Binkley, J. A. Pople, and W. J. Hehre, J. Am. Chem. Soc. 102, 939 (1980). 

103. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry. Introduction to Advanced 
Electronic Structure Theory, McGraw-Hill, New York, 1989. 

104. P. Jorgensen and J. Simons, Second Quantization-Based Methods in Quantum 
Chemistry, Academic Press, New York, 1981. 

105. R. McWeeny, Methods of Molecular Quantum Mechanics, Academic Press, New 
York, 2nd edition, 1992. 

106. A. E. Dorigo and P. von R. Schleyer, Chem. Phys. Lett. 186, 363 (1991). 

107. R. F. W. Bader, W. H. Henmeker, and P. E. Cade, J. Chem. Phys. 46, 3341 (1967). 

108. T. Clark, J. Chandrasekhar, G. W. Spitznagel, and P. v. R. Schleyer, J. Comput. 
Chem. 4, 294 (1983). 



BIOGRAPHICAL SKETCH 



Ricardo Longo was bom in Sao Paulo, Brazil on June 11, 1964, to Dalva and 
Adhemar Longo. He has a sister Renata and a brother Milton, both younger than he. 

As a kid, in Pen^polis, he used to mix chemicals and make experiments. A chemist 
was being bom. However, after reading a Portuguese translation of Gamow’s ’The 
Great Physicists, from Galileo to Einstein’ he was very enthusiastic about physics, so 
he became a physical-chemist. In August, 1982 he enrolled in chemistry at Federal 
University of Sao Carlos. During his chemistry major in the Department of Chemistry at 
Sao Carlos, he received several scholarships from the Brazilian government for scientific 
research and teaching. As a result, he worked in organic synthesis, electrochemistry, 
high vacuum pressure gauges, and quantum chemistry. He also started a physics major 
in the Physics Department at University of Sao Paulo, at Sao Carlos campus. He did 
not have time to graduate from the latter. After getting his BA in chemistry he almost 
gave up quantum chemistry, when he had the opportunity to perform his master degree’s 
research at Fundamental Chemistry Department at Federal University of Pernambuco, in 
Recife. There he met Michael Zemer and made up his mind to come to Gainesville to 
get his degree in the Quantum Theory Project. So, after getting his master’s degree on 
May, 1988 he was granted a scholarship from the Coordenadoria de Aperfei 9 oamento de 
Pessoal de Ensino Superior (CAPES) and came to Gator’s Country to start working on 
his doctoral degree in the Fall of 1988. 

He hopes to start a research group in Brazil, perhaps in a good university, and make 
some contributions to science and society. 



129 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 




N. Yngve 0hm, Chairman 
Professor ot Chemistry 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 




Michael C. Zemer 
Professor of Chemistry 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 

// I /I . 

. If . 

William Weltner, Jr. 

Professor of Chemistry 





I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 




Graduate Research Professor of 
Chemistry 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



Professor of Chemical 
Engineering 




This dissertation was submitted to the Graduate Faculty of the Department of 
Chemistry in the College of Liberal Arts and Sciences and to the Graduate School 
and was accepted as partial fulfillment of the requirements for the degree of Doctor 
of Philosophy. 



May 1993 

Dean, Graduate School