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Time-dependent theoretical treatments of the dynamics
of electrons and nuclei in molecular systems
E. Deumens, A. Diz, R. Longo, and Y. Ohrn
Quantum Theory Project University of Florida, Gainesville, Florida 32611
An overview is presented of methods for time-dependent treatments of molecules as systems of electrons
and nuclei. The theoretical details of these methods are reviewed and contrasted in the light of a recently
developed time-dependent me thod called electron-nuclear dynamics. Electron-nuclear dynamics (END) is
a formulation of the complete dynamics of electrons and nuclei of a molecular system that eliminates the
necessity of const ructin g potential-ener gy surfaces. Because of its general formulation, it encom passes
many aspects found in other formulations and can serve as a didactic device for clarifying many of the
principles and approximations relevant in time-dependent treatments of molecular systems. The END
equations are derived from the time-dependent variational principle applied to a chosen family of
efficiently parametrized approximate state vectors. A detailed analysis of the END equations is given for
the case of a single-determin antal sta te for the electrons and a classical treatme nt of the nuclei. The ap
proach leads to a simple formulation of the fully nonlinear time-dependent Hartree-Fock theory including
nuclear dynamics. The nonlinear END equations with the ab initio Coulomb Hamiltonian have been im
plemented at this level of theory in a computer program, ENDyne, and have been shown feasible for the
study of small molecular systems. Implementation of the Austin Model 1 semiempirical Hamiltonian is
discussed as a route to large molecular systems. The linearized E N D equatio ns at this level of theory are
shown to lead to the random-phase ap proximation for the coupled system of electrons and nuclei. The
qualitative features of the general nonlinear solution are analyzed using the results of the linearized equa
tions as a first approximation. Some specific applications of EN D are presented, and the comparison with
experiment and othe r theoretical approaches is discussed.
CONTENTS
I. Introduction
A. Plan of presentation
B.
Overview of methods
1.
Potential-energy surfaces
2. Quantum dynamics on a single surface
a. Exact wave-packet propagation
b.
Time-dependent self-consistent field
3 . Trajectories on a single surface
a. Fitted surface
b.
Computed surface
4. Dynam ics of electrons and nuclei
a. Wave-packet propagation on coupled potential
surfaces
b.
Trajectories on coupled potential surfaces
c. Car-Parrinello method
d. Time-dependent Hartree-Fock
e. Time-dependent density functional
f. Close coupling and perturbed stationary state
g. Electron-nuclear dynamics
II . Preparations
A. Electronic spin-orbitals
1. Choice of basis and convergence
2.
Atomic spin-orbitals
3 . Molecular spin-orbitals
4.
Electron translation factors
B.
Electronic wave function
1.
Dynamic orbitals
2.
Coherent states and Lie groups
C. Treatm ent of the nuclei
D .
The time-dependent variational principle
I I I .
Electron-Nuclear Dynamics
A. Orthon ormal representation
1.
Representation in an orthono rmal basis fixed in
space
2 . Symplectic transformation to traveling atomic or
bitals
3. Representation in an orthonormal basis built with
traveling atomic orbitals
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940
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IV.
V.
B.
C.
Nonorthogonal representation
1.
Symplectic transformation to raw atomic orbitals
2.
Representation in the nonorthogonal basis of trav
eling atomic orbitals
Analysis
1. Molecules in inertial motion
a. Using traveling atomic orbitals
b. Using orbitals fixed in space
2.
Linearized equations
3 .
Equation for the density
4. Velocity-dependent terms
5.
Acceleration-dependent terms
Applications
A .
B.
C.
D .
Implementation
Ion-atom scattering
1.
Proton-hydrogen collisions
2. Proton-helium collisions
Ion-molecule scattering
Intramolecular electron transfer
Discussion
A.
B.
Invariance principles
1.
Translation invariance in time
2. Translation invariance in space
3 . Rotational invariance
4. Time-reversal invariance
Conclusion
Acknowledgments
Appendix
1.
2.
3.
Derivation in an orthonormal basis
a. Metric
b. Density matrix
c. Energy
Derivation in the atomic-orbital basis
a. Definition of parameters
b.
Dynamic orbitals
c. Metric
d. Density matrix
e. Energy
Details of semiempirical approaches
References
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Reviews of M odern Physics, Vol. 66, No. 3, July 1994
0034-6861
/94/66(3)/917(67 )/$11.70
1994
The American Physical Society
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I. INTRODUCTION
Dy n am ics o f m o lecu l a r sy s t em s h as b een o f i n t e res t i n
ch em ica l p h y s i cs an d p h y s i ca l ch em is t ry s i n ce t h e ad v en t
o f q u a n t u m m e c h a n i c s . O v e r t h e y e a r s m a n y t h e o r e t i c a l
ap p ro ac h es h av e b een fo rm u la t ed an d im p lem e n ted i n
d e t a i l . Co m p u ta t i o n a l m eth o d s fo r e f f i c i en t so lu t i o n o f
th e r esu l t i n g eq u a t io n s h av e b een d ev i sed i n m an y cases .
Al th o u g h th e b as i c eq u a t io n o f q u an tu m m ech an ics i s a
t im e-d ep en d en t eq u a t io n , i n t h e Sch ro d in g er , He i sen -
b erg , o r i n t e rac t i o n r ep resen t a t i o n , t h e f ac t t h a t sep ara
t ion of var iab les i s possib le and the in t ractab i l i ty of the
equat ion for systems of any complexi ty have led to a s i tu
a t i o n i n wh ich sev era l g en era t i o n s o f sc i en t i s t s h av e b een
t r a in ed t o fo cu s o n t h e t im e- in d ep en d en t eq u a t io n b o th
fo r b o u n d - s t a t e an d fo r sca t t e r i n g p ro b l em s . In t h e la s t
d eca d e , h o we v er , t h e t im e-d ep en d en t fo rm u la t i o n h as a t
t r ac t e d m o re a t t en t i o n . I t h as b eco m e c l ea r t h a t so lv in g
th e t im e-d ep en d en t eq u a t io n d i r ec t l y is i n d eed p o ss ib l e
an d so m et im es co m p u ta t i o n a l l y ad v an tag eo u s o v er f i r s t
f ind ing al l relev ant s tat io nar y-s tate eigenfu nct ions for a
g iv en p ro b l em .
T im e-d ep en d en t d esc r ip t i o n s h av e a lway s ap p ea l ed t o
research er s ' i n tu i t i o n , a s ev id en ced b y t h e f ac t t h a t ex
p er im en t s a r e o f t en d i scu ssed , b o th b y ex p er im en te r s an d
b y th eo r i s t s , i n a t im e-d ep en d en t l an g u a g e . On e o f t h e
m o re d ram at i c s i t u a t i o n s i n t h i s r esp ec t is p ro v id e d b y
th e f ie ld o f f em to seco n d sp ec t ro sco p y (Zewai l an d Bern
stein , 1988; Zew ai l , 1989) wh ere the t im e evolu t ion of nu
clei and elect rons in molecular systems can be fo l lowed in
d e t a i l . I n t h e l a s t d eca d e , t im e-d ep en d en t t h eo re t i ca l
t r ea tm en t s h av e m atu red t o t h e ex t en t t h a t d e t a i l ed
t h e o r e t i c a l d e s c r i p t i o n s m a t c h i n g e x p e r i m e n t a l a c c o m
p l i sh m en t s a r e f eas ib l e . An o th er i n d i ca to r o f t h e g ro w
in g im p o r t an ce o f t im e- d ep e n d en t d esc r ip t i o n s i s t h e
d i sco v ery , i n t h e t h eo ry o f m easu rem en t , t h a t h i s t o r i es o f
ev en t s as t h e b as i c co n ce p t , r a th e r t h an t h e ev en t s t h em
se lv es , l ead t o a co n s i s t en t i n t e rp re t a t i o n o f q u a n tu m
m e ch an ics . In an im p o r t a n t r ev i ew, Om n es (19 9 2 ) sh o ws
th a t co n s id era t i o n o f co n s i s t en t seq u en ces o f p ro p er t i e s
in t ime is an essen t ial ingred ien t and al lows the
Co p en h ag en In t e rp re t a t i o n t o b e ex t en d ed t o a co n s i s t en t
i n t e rp re t a t i o n o f q u a n tu m m ec h an ic s . I t i s t o o ea r ly t o
s t a t e t h a t t h ese d ev e lo p m en t s wi l l en d t h e l o n g d eb a t e o n
th e i n t e rp re t a t i o n o f q u an tu m m ech an ics ; b u t i t i s n o t
ab le that expl ici t considerat ion of t ime is pu t for th as a
k ey e l em en t i n t h a t l o n g d eb a t e , an d t h a t i t s i n t ro d u c t io n
actual ly s t rengthens and s impl i f ies the or ig inal
C o p e n h a g e n I n t e r p r e t a t i o n .
Th e a im s o f t h i s p ap er a r e (1 ) t o r ev i ew th e m o s t p ro m
in en t o f t h e t im e-d ep en d en t m eth o d s , an a ly z in g an d co m
p ar in g t h e p r in c ip l es an d ap p ro x im at io n s u n d er ly in g
th em as we l l a s co m m en t in g o n t h e i r co m p u ta t i o n a l im
p l em en ta t i o n ; (2) t o p resen t m o s t o f t h ese m eth o d s an d
techniques f rom a general , un i f ied , and d idact ic po in t o f
v i ew, wh ic h is p ro v id ed b y t h e t im e-d e p en d e n t v a r i a t i o n
al p r inc ip le; and (3) to ou t l ine in conside rab le d etai l on e
m e t h o d e l e c t r o n - n u c l e a r d y n a m i c s ( E N D ) t h a t i s a
r i g o ro u s ap p l i ca t i o n o f t h e t im e-d ep e n d en t v a r i a t i o n a l
pr inc ip le and is there fore a usefu l f ram ew ork for exp lain
in g t h e di f fe ren ces b e tween v ar io u s m eth o d s . Th e m a in
em p h as i s is o n t im e-d ep en d en t m e th o d s ; b u t b e cau se
p o ten t i a l - en erg y su r faces p l ay su ch an im p o r t an t ro l e i n
th e t h eo ry o f m o le cu l a r p h en o m en a , so m e a t t en t i o n i s
p a id t o t h e co n s t ru c t i o n o f ap p ro x im at io n s t o e ig en s t a t es
o f t h e e l ec t ro n i c Ha m i l to n i an . Ho w ev er , n o a t t e m p t i s
m ad e t o r ev i ew th e f i e ld o f e l ec t ro n i c s t ru c tu re t h eo ry .
On ly t h o se m eth o d s a re m en t io n ed t h a t i n so m e way can
b e co n s id ered as sp ec i a l cases o f t im e-d ep en d en t
m e t h o d s .
Al though we t ry to g ive a fai r account of the s tate of
the f ield , our rev iew is b iased by our ow n exper ien ces an d
i s n o t i n t en d ed t o b e ex h au s t i v e . W e ap o lo g i ze fo r an y
o m iss io n s .
Ato m ic u n i t s wi l l b e u sed t h ro u g h o u t t h i s wo rk ; i . e . ,
ft= 1, m
e
= 1 , and e1 .
A. Plan of presentation
To p resen t an o v erv i ew o f t h e m an y th eo re t i ca l
m e t h o d s o f t im e - d e p e n d e n t t r e a t m e n t s o f m o l e c u l a r p r o
cesses , (i ) the t ime -de pen den t var iat io nal p r inc ip le and ( i i)
t h e co n cep t o f p o t en t i a l - en erg y su r faces a re u sed .
Th e t im e-d ep en d en t v a r i a t i o n a l p r in c ip l e (TDVP) ,
wh en th e t r i a l wav e fu n c t io n i s co m p le t e ly g en era l an d
n o t r es t r i c t ed i n an y fo rm , y i e ld s t h e t im e-d ep en d en t
Sch ro d in g er eq u a t io n (Di rac , 1 9 30 ) , a s sh o wn in Sec .
I I . D .
On th e o th er h an d , an y ch o i ce o f a r es t r i c t ed fo rm
o f t r i a l fu n c t i o n i n t h e TDVP resu l t s i n t im e-d ep en d en t
eq u a t io n s t h a t ap p ro x im ate t h e Sch ro d in g er eq u a t io n .
Based on th is , one can d iscern three d i f feren t classes of
m e t h o d s . F i r s t , t h e ti m e - d e p e n d e n t S c h r o d i n g e r e q u a
t i o n ( in p r in c ip l e o b t a in e d f ro m th e co m p le t e ly u n re
s t r i c t ed TDVP) can b e s tu d i ed an d ap p ro x im ate so lu t i o n s
fo u n d to o b t a in t r an s i t i o n am p l i t u d es an d sp ec t r a l i n fo r
m a t io n i n a v a r i e ty o f way s . Seco n d , a sep ara t i o n o f v a r i
ab l es y i e ld s t h e t im e- in d ep en d e n t Sch ro d in g er eq u a t io n ,
wh ich can b e s tu d i ed t o f i n d ap p ro x im ate s t a t i o n ary - s t a t e
so lu t i o n s . Th e t h i rd c l as s o f m e th o d s a re t h o se , a s s t a t ed
ab o v e , t h a t d i r ec t l y r es t r i c t t h e TDVP t r i a l fu n c t i o n an d
o b t a i n d y n a m i c a l e q u a t i o n s t h a t a p p r o x i m a t e t h e t i m e -
d ep en d en t Sch ro d in g er eq u a t io n .
A l t e r n a t i v e v a r i a t i o n a l p r i n c i p l e s k n o w n a s t h e D i r a c -
F ren k e l v a r i a t i o n a l p r in c ip l e (Di rac , 1 9 30 ; F ren k e l , 1 934 )
a n d t h e M c L a c h l a n v a r i a t i o n a l p ri n c i p l e ( M c L a c h l a n
and Bal l , 1964) are equivalen t to the TDVP as long as
co m p lex wav e fu n c t io n s an a ly t i c i n t h e p a ram ete r s a r e
used and the g lobal phase of the wave funct ion i s retained
( K u c a r e * ah , 1 9 8 7 ; Bro eck h o v e et ah, 1988).
Ear l i e r t h eo re t i ca l wo rk r e l a t ed t o t h e g en era l ap
p ro a ch o f t h i s r ev i ew is t h a t o f Ker m an an d Ko o n in
(1976), w ho f irst show ed tha t the TD V P appl ie d to a
p ara m et r i zed f am i ly o f wav e fu n c t io n s l ead s t o a c l as s ica l
Ha m i l to n i an sy s t em o f eq u a t io n s fo r t h e p a ra m e te r s .
Ro w e an d Basse rm an (1 97 6 ) l a t e r i n t ro d u ce d t h e t h eo r y
o f co h e ren t s t a t es t o p ro v id e a g en era l f r am ew o rk fo r
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p a r a m e t r i z e d w a v e f u n c t io n s , a n d K r a m e r a n d S a r a c e n o
(1 9 8 1 ) wo rk ed o u t t h e g eo m et ry o f t h e TDVP.
Tre a tm en t s o f m o le cu l a r sy s t em s o ft en u se t h e co n cep t
of po ten t ial -energy surface ob tained (at least in pr incip le)
f ro m so lv in g t h e t im e- in d ep en d en t Sch ro d in g er eq u a t io n
wi th t h e e l ec t ro n i c Ham i l to n i an fo r a l a rg e n u m b er o f
f i x ed n u c l ea r g eo m et r i es . Th e wid e ly u sed co n cep t o f a
p o t en t i a l - en erg y su r face can se rv e t o d i s t i n g u i sh b e tween
d i f f e ren t ap p ro ach es , i n t h a t i t su m m ar i zes h o w th e e l ec
t ro n i c su b sy s t em i s t r ea t ed .
Meth o d s t h a t u se a s i n g l e p o t en t i a l - en erg y su r face
a d h e r e t o t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n , o r t h e
TA BL E I. Overview of theore tical molec ular method s classified by the use of potential-energy surface and time dependen ce. The na-
ture of the treatment of the nuclei is indicated as classical, semiclassical, and quantum mechanical.
PES Nuclei
Time independent Time dependent
Single
Classical
Semiclassical
Quantum
Multiple
Not used
Classical
Semiclassical
Quantum
Classical
Semiclassical
Quantum
Energetics, equilibrium
geometries, transition
states, barrier heights,
reaction paths
Reaction rates from
transition-state theory
(Melissas et ah, 1992)
Vibrational and rotation
al eigenstates
(Lathouwers
et al.,
1987),
multichannel scattering
and reactive collisions
with or without con
strained geometries,
(variational) R-matrix
(Linderberg et al., 1989)
an d
5-matrix
(Miller and
op de Haar, 1987) ap
proaches, method of per
turbed stationary states
(Mott and Massey, 1965;
Riera, 1992)
Reactive processes and
calculation of nonadia-
batic couplings (Lengfield
and Yarkony, 1992)
Diagonalization in a
basis of electronic and
vibrational states (Kolos
and Wolniewicz, 1964),
many-body scattering
calculations as basis for
molecular structure cal
culations (Levin, 1978)
and for molecular reac
tions (Micha, 1985)
Molecular dynamics us
ing model forces or the
gradient of a fitted or
directly computed PES
(Carmer et al., 1993)
Semiclassical molecular
dynamics, t ime correla
tion functions in eikonal
approximation (Micha
and Villalonga, 1993),
wave-packet dynamics
(Huber
et al.,
1988)
Quantum molecular dy
namics with representa
tion on a grid of points
or with a set of basis
functions (Feit and
Fleck, 1980; KoslofF and
Kosloff, 1983a; Tal-Ezer
an d Kosloff, 1984;
Leforestier et al., 1991;
Man th e
et al.,
1992b),
single and
multiconfiguration time-
dependent self-consistent
field (Manthe et al.,
1992a)
Surface hopping model
(Tully and Preston, 1971)
Wave-packet dynamics
on vibronically coupled
surfaces (Coalson, 1989;
Manthe et al., 1991)
Time-dependent Hartree-
Fock (Dirac, 1930)
Car-Parrinello (1985),
electron-nuclear dynam
ics (Deumens, Diz, Tay
lor, and Ohrn, 1992),
close-coupling methods
(Delos, 1981)
Perturbed-stationary-
states and close-coupling
methods (Delos, 1981)
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Deumens et al.: Dynam ics of electrons and nuclei
ad i ab a t i c ap p ro x im at io n i f t h e d i ag o n a l co r rec t i o n t e rm
to t h e n u c l ea r k in e t i c en erg y
(q?(R)\A
R
\
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Dynamics of electrons and nuclei
921
m o le cu l a r p o t en t i a l - en erg y su r face (PES) wi th an as so c i
a t ed e l ec t ro n i c e ig en s t a t e . Th e m eth o d s o f co n s t ru c t i o n
of rel iab le PES's const i tu te an ex tensive f ield of s tudy
( see , e .g . , t h e r ev i ew b y Du n n in g an d Hard in g , 1 9 8 5 , fo r
ab initio su r f aces an d t h e r ev i ew b y Ku n tz , 1 9 8 5 , fo r sem -
i em p i r i ca l o n es ) . Th e d y n am ics o f t h e m o lecu l e i s t h en
red u ced t o t h e d y n am ics o f t h e n u c l e i o n t h a t PES , a t t h e
c l as s i ca l , s em ic l as s i ca l , o r q u an tu m l ev e l . T h i s ap p ro a ch
raises the fo l lowing issues that are addressed in d i f feren t
way s b y t h e v ar io u s m eth o d s .
(1 ) Co n s t ru c t i o n o f t h e e l ec t ro n i c e ig en s t a t es fo r o n e
g eo m et ry i s a n o n t r i v i a l p ro b l em in itself. In o rd er t o o b
tain a usefu l repre sen tat i on of the surface for al l
g eo m et r i es n eed ed fo r t h e r e l ev an t d y n am ics , o n e m u s t
r eso r t t o so lv in g t h e e l ec t ro n i c p ro b l em a t se l ec t ed
g eo m et r i es an d p er fo rm an i n t e rp o l a t i o n o r f i t u s in g a
su i tab le analy t ic form for the surface to def ine i t s value at
o th er g eo m et r i es .
(2) Th e proc ess of con st ru ct in g a fai th fu l f i t i s a lso a
sizab le task . Us ual ly d if feren t ana ly t ic forms are needed
for d i f feren t reg ions. A nd i t has been found (Liu an d
Mu r re l l , 1 9 9 1 ; Ag u ad o an d Pan i ag u a , 1 9 9 2) t h a t sm al l
errors in curvature of the f i t can produce s ign i f ican t ly
d i ff e rent d y n am ics . Ac cu r a t e f it s a r e av a i l ab l e fo r so m e
t r i a to m ic sy s t em s an d fo r a f ew fo u r -a to m m o lecu l es . I t
i s general ly bel ieved that f i t s for general po lyatomic sys
t em s wi l l b e h a rd t o co m e b y .
(3) A n e l ec t ro n i c s t a t e as so c i a t ed w i th a PES i s co m
p le t e ly s t a t i c . Th e e r ro r s i n t ro d u ced i n t h e m o lecu l a r d y
n am ics a re u su a l l y n eg l ec t ed b y i n v es t i g a to r s l o o k in g fo r
l o w - e n e r g y a n d a d i a b a t i c r e a r r a n g e m e n t r e a c t i o n s a n d
in f ra red sp ec t ro sco p y . Ho w ev er , fo r o th e r s i t u a t i o n s ,
su ch as ch arg e- t r an s fe r r eac t i o n s tu d i es , so m e o f t h e e r
ro r s can n o t b e n eg l ec t ed . To o v erco m e th a t l im i t a t i o n ,
m eth o d s su ch as p e r tu rb ed s t a t i o n ary s t a t es u se m u l t i p l e
PES ' s , an d c lo se -co u p l in g o r t im e-d ep en d en t Har t r ee -
Fo c k m eth o d s u se e l ec t ro n t r an s l a t i o n fac to r s (ET F ' s ) .
Th e m eth o d s t h a t r e ly o n a s i n g l e o r m u l t i p l e PES as
su m e t h a t a su r f ace i s g iv en i n a n u m er i ca l l y access ib l e
fo rm . So m et im es t h i s is acco m p l i sh ed w i th a g lo b a l fit o r
patches of local sp l ine f i t s to a set o f po in ts ob tained f rom
accu ra t e e l ec t ro n i c s t ru c tu re ca l cu l a t i o n s . Th e b es t
m eth o d s fo r g lo b a l su r f aces a re e l ec t ro n i c
mul t iconf igurat ional sel f -consis ten t - f ield calcu lat ions (Ol-
se n
et aL,
1 9 8 3 ; Jen sen an d Ag r en , 1 9 8 6 ) . Fu l l
conf igurat ion in teract ion wi th a real i s t ic basis i s , o f
course, feasib le on ly for the smal lest systems. A sys
t em at i c p ro ced u re fo r g en era t i n g m o d e l p o t en t i a l - en erg y
su r faces fo r g en era l m o lecu l es i s t h a t o f d i a to m ics i n m o l
ecu l es , wi th ab initio a n d s e m i e m p i ri c a l i m p l e m e n t a t i o n s
(Du n n in g an d Hard in g , 1 9 8 5 ) .
Meth o d s fo r co n s t ru c t i n g t h e PES a t t h e sam p l in g
p o in t s , wh ich u su a l l y i n c lu d e c r i t i ca l p o in t s l i k e l o ca l
m in im a , sad d l e p o in t s , an d r eac t i o n p a th s , a r e n o t
co v ered b y t h i s r ev i ew; n e i t h e r a r e t h e t ech n iq u es o f i n
t e rp o l a t i o n o f PE S d a t a p o in t s . Ho w ev er , so m e of t h e
t im e-d ep en d en t m eth o d s , wh ich d o n o t r e ly o n t h e PES ,
p o ssess a n a tu ra l t im e- in d ep en d en t sp ec i a l case t h a t can
effect ively be used to f ind eigenstates to the elect ron ic
Ham i l to n i an an d , t h u s , t o co n s t ru c t PES ' s ( see Sec .
I I I .C) .
2.
Quantum dynamics on a single surface
Th ere i s a g ro win g l i t e r a tu re o f m eth o d s t h a t can b e
r e f e rr e d t o a s " d i r e c t a p p r o a c h e s . " S u c h n u m e r i c a l
m e th o d s h av e two k ey f ea tu res . On e i s t h e r ep resen t a
t ion of the wav e funct io n , ei th er by expa nsion coeff icien ts
in a basis set o r by d iscret izat ion on a gr id of po in ts .
T h i s ,
i n t u rn , d e t e rm in es h o w to ev a lu a t e t h e ac t i o n o f
t h e o p e r a t o r s , i n p a r t i c u l a r , t h e H a m i l t o n i a n ft= + P
wi th i t s k in e t i c - en erg y o p era to r R an d i t s p o t en t i a l -
en erg y o p era to r P , o n t h e wav e fu n c t io n . Th e o th e r
fea tu re i s t h e ac tu a l t im e p ro p ag a t io n a lg o r i t h m , i . e .,
h o w th e so lu t i o n i s m a rch ed o v e r a t im e in t e rv a l , g iv en
i ts value at the in i t ia l t ime.
a. Exact wave-packet propagation
I t i s p o ss ib l e t o so lv e ex ac t l y t h e Sch ro d in g er eq u a t io n
fo r t h e i n t e rn a l d eg rees o f f r eed o m o f m o lecu l es wi th
th ree o r fo u r a to m s b y r ep resen t in g t h e wav e fu n c t io n o n
a n u m er i ca l g r id . Th ese m eth o d s a re k n o wn as t h e
d i sc re t e v a r i ab l e r ep resen t a t i o n an d t h e p seu d o sp ec t r a l
ap p ro x im at io n , d ep en d in g o n wh e th er o n e p re fe r s t o
s t r es s t h e n u m er i ca l r ep resen t a t i o n o r t h e m eth o d o f
s o l u t io n . T h e p s e u d o s p e c t r a l F o u r i e r a p p r o x i m a t i o n
(Go t t l i eb an d Orszag , 1 9 77 ) was i n t ro d u ced r a th e r r e
cen t ly i n to m o lecu l a r d y n am ics , a l t h o u g h fo r so m e t im e
i t had been used in several areas of physics , fo r example,
f lu id d y n a m ic s , o p t i cs , an d e l ec t ro n m icro sco p y . Th i s
ap p ro ach em p lo y s a g r id r ep resen t a t i o n o f t h e wav e fu n c
t i o n i n co o rd in a t e sp ace an d ap p l i es t h e d i sc re t e Fo u r i e r
t r a n s f o r m t o o b t a i n a m o m e n t u m s p a c e r e p r e s e n t a t i o n .
Co m p u te r co d es p e r fo rm in g fas t Fo u r i e r t r an s fo rm a re
general ly avai lab le, making i t possib le to swi tch f rom
co o rd in a t e sp ace t o m o m en tu m sp ace an d b ack so as t o
al low for fas t evaluat ion of the act ion of the k inet ic-
en erg y an d t h e p o t en t i a l - en e rg y t e rm s o f t h e H am i l to n i
an . E r ro r an a ly s i s an d co l l o ca t i n g fu n c t io n s o th er t h an
p lan e wav es a re av a i l ab l e wi th in t h i s t ech n iq u e . A p
propr iate funct ions for rad ial (Bissel ing and KoslofF,
1 98 5) an d an g u la r co o rd in a t es (Q u ere an d Lefo res t i e r ,
1 9 9 0 ) h av e b een im p lem en ted .
T h e r e a r e v a r i o u s t i m e i n t e g r a t i o n a l g o r i t h m s c u r r e n t
l y i n u se wi th t h e p seu d o sp ec t r a l m eth o d in m o lecu l a r
d y n am ics . On e can d i s t i n g u i sh fo u r , n am ely , t h e
seco n d -o rd er -d i f fe ren ce m e th o d , t h e sp l i t -o p era to r
m e t h o d , t h e s h o r t - t i m e i t e r a t i v e L a n c z o s m e t h o d , a n d
t h e C h e b y s h e v e x p a n s i o n m e t h o d .
Wh en th e seco n d -o rd er d i f f e ren ce (Ko s lo f f an d Kosloff,
1 9 8 3a) i s ap p l i ed t o t h e Sch ro d in g er eq u a t io n , t h e wav e
fu n c t io n i s co m p u te d a t su ccess iv e sh o r t t im e s t ep s
t h r o u g h a c o m b i n a t i o n o f a f o r w a r d a n d a b a c k w a r d
m o v e , wh ich e l im in a t es t h e seco n d -o rd er t e rm s ; so t h e e r -
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ro r i s o f t h i rd o rd er i n t h e t im e s t ep . Th e o th er t h ree
m e th o d s fo cu s o n t h e p ro p ag a to r . In t h e sp l i t -o p era to r
sch em e (Fe i t an d F l ec k , 1 9 8 0 ) , o n e m ak e s a sy m m et r i c
d eco m p o s i t i o n o f t h e ex ac t p ro p ag a to r
6 )
= exp -ifie) ,
(1.1)
1 7 (e ) ex p
i *
ex p ( /eK)ex p
e ^
i n t ro d u c in g a t h i rd - o rd e r e r ro r t e rm . Th i s i s a sh o r t - t im e
m eth o d th a t r eq u i r es su ccess iv e ap p l i ca t i o n s t o p ro p a
gate over a f in i te t ime in terval .
Th e sh o r t - t im e i t e r a t i v e Lan czo s p ro p ag a t io n fo rm u la
(Leforest ier et al., 1991) is
l / ( 6 ) e x p [ - / A ( f T , ^ ( 0 ) ) ]
(1.2)
w h e r e Ui s n o w a m at r i x o p era to r i n t h e Kry lo v su b sp ac e
g en era t ed b y t h e Ham i l to n i an an d t h e i n i t i a l wav e fu n c
t i o n . Th e m at r i x A i s t h e tr i d i ag o n a l Lan c zo s m a t r i x
rep resen t in g t h e Ham i l to n i an i n t h e Kry lo v sp ace (Cu l -
l u m an d Wi l lo u g h b y , 1 9 8 5 ). Th e ex p o n en t i a t i o n i s u su a l
l y p e r fo rm ed b y d i ag o n a l i z in g t h e Lan czo s m at r i x an d
wo rk in g wi th t h e d i ag o n a l e ig en v a lu e m at r i x . Th e l en g th
of t ime d ictates the s ize of the Krylov space needed for a
predef ined accuracy (Park and Light , 1986) . In general ,
shor t t ime s teps are used in order no t to lose the advan
t ag e o f t h e Lan czo s r ed u c t io n ; i . e . , i t e r a t i n g t h e a lg o
r i t hm is mo re efficien t tha n using long t im e s teps .
Th e Ch eb y sh ev ex p an s io n m eth o d (Ta l -Ezer an d
Kosloff, 1 98 4 ) ap p r o x im ates t h e ex ac t p ro p a g a to r b y a
Ch eb y sh ev ex p an s io n ,
# ( f ) = 2
a
n
(t)T
n
{-iH
R
)
(1.3)
wh ere t h e Ham i l to n i an n eed s t o b e r en o rm al i zed so t h a t
i t s sp ec t ru m co in c id es wi th t h e d o m ain o f t h e Ch eb y sh ev
p o l y n o m i a l s T
n
. Th i s i s a l o n g - t im e m e th o d . Th e co n
v erg en ce r eq u i r em en t o n t h e Ch eb y sh ev ex p an s io n i s
su ch t h a t t h e n u m b er o f t e rm s d o es n o t d ecrease
signi f ican t ly for smal ler t value s . Th us , for eff iciency , t
sh o u ld b e l a rg e .
Sh o u ld o n e wan t t o co n s id er t im e-d ep en d en t Ham i l -
t o n i an s , t h e re a re s t r a ig h t fo rward way s t o ex t en d t h e
sp l i t -o p era to r , s eco n d -o rd er -d if f e ren ce , an d sh o r t - t im e
i t e ra t i v e Lan czo s m eth o d s t o su ch cases , wh i l e t h e Ch e
b y sh ev ex p an s io n m eth o d wo u ld seem n o t t o h av e t h i s
f lexibi li ty . Fo r sh o r t - t im e m e th o d s , fu r th e r ap p r o x im a
t i o n s a re u su a l l y im p l i ed , su ch as t h e u se o f sh o r t - t im e
av erag e d Ham i l to n i a n s an d t h e d i s r eg ard of t im e o rd e r
ing .
Th e sp ec i f i c m er i t s o f v a r io u s m eth o d s t o ev a lu a t e t h e
ac t i o n o f t h e Ham i l to n i an o n t h e wav e fu n c t io n an d t o
p ro p ag a t e t h e so lu t i o n i n t im e h av e b een r ev i ewed b y
Kosloff (1988) . The computat ional effor t involved in
th ese m eth o d s i s p resen t ly su ch t h a t ap p l i ca t i o n s i n v o lv
in g a t m o s t a f ew d eg rees o f f r eed o m can b e a t t em p ted .
Th ey a re u sed t o s t u d y q u an tu m d y n am ics o f n u c l ea r
m o t io n o n a p o t en t i a l - en erg y su r face o r o n a sm al l n u m
b er o f co u p led p o t en t i a l - en erg y su r faces . Th e sp l i t o p era
tor offers l imi ted accuracy and has been used in s tud ies of
react ions (Kosloff and Kosloff, 1 9 83b ) an d ab so rp t i o n
sp ec t r a (Tan g et al., 1990) . Th e sec ond -ord er d i f ference
has been used ex tensively because of i t s ease of im
p l em en ta t i o n o f e ig en sp ec t r a (Fe i t
et al.
9
1982) , nonadia-
b a t i ca l l y co u p led sy s t em s (Alv are l l o s an d Met iu , 1 9 8 8 ;
M an th e an d K o p p e l , 1 9 9 0 b ), d i s so c i a t i o n an d p red i s so c i -
a t i o n p ro cesses (Man th e et ah, 1 9 9 1) , p h o to d e t ac h m en t
sp ec t r a (En g e l , 1 9 91 ) , an d sy s t em s wi th t im e-d e p en d e n t
H a m i l t o n i a n s ( C h e s l o w s k i et al., 1990). A significant
n u m b er o f ap p l i ca t i o n s of t h e Ch eb y sh e v ex p an s io n
m e t h o d h a v e b e e n m a d e , a m o n g o t h e r p r o b l e m s , t o
a to m -d ia to m co l l i s i o n s (Su n
et al.,
1987), to a
m u l t i co n f ig u ra t i o n a l se l f - co n s i s t en t - f i e ld ap p ro ach (Ham -
m e r i c h et al., 1 9 9 0) , t o p h o to d i s so c i a t i o n (Ku la n d er
et al., 1991), and to the co mp uta t io n of energy levels
(Kosloff and Tal -E zer , 1986; N eu ha use r , 1990) .
b. Time-dependent self-consistent field
Th i s m eth o d ex p lo red b y Ko ss lo f f an d Ra tn er (B i s se l -
in g et al., 1 9 8 7) , am o n g o th e r s , r ed u ce s t h e m a n y v ar i
ab le Schrodinger equat ions for the nuclei to a set o f cou
p l ed eq u a t io n s fo r each n u c l ea r co o rd in a t e m o v in g i n t h e
av erag e f ie ld of t h e o th er s . W av e p ack e t s a r e r ep rese n t ed
o n a g r id an d a re p ro d u c t s o f p ack e t s , each i n o n e i n t e r
val co ord ina te on ly . S ince a s ing le conf igu rat ion s tate
can n o t p ro p er ly d esc r ib e t h e d iv i s io n o f p ro b ab i l i t y o v er
two ch an n e l s , t h e m eth o d was ex t en d ed t o a l l o w fo r
m u l t i co n f ig u ra t i o n a l s t a t es . T h i s p e rm i t s a p ro p er
d esc r ip t i o n o f r eac t i o n s wh ere two ex i t ch an n e l s a r e p o p
u l a t ed . Th e m e th o d i s a l so k n o wn as t h e t im e-d ep e n d en t
H a r t r e e a p p r o x i m a t i o n . A v a r i a ti o n o n t h e m e t h o d ,
ca l l ed t im e-d ep en d en t ro t a t ed Har t r ee , was d ev e lo p ed b y
M e y e r a n d c o - w o r k e r s ( M e y e r et al., 1988).
3. Trajectories on a single surface
a. Fitted surface
M o l e c u l a r d y n a m i c s a s t h e N e w t o n i a n m e c h a n i c s o f
nuclei wi th effect ive two - , th re e- , and four-body forces
h as b een u sed wi th g rea t su ccess t o s t u d y m an y m o lecu
lar p rocess es . Fo r large molecule s th is i s s t i l l the on ly
pract ical method , and i t i s in general use, especial ly in or
g an i c ch em is t ry an d b io ch em is t ry . I t is u sed t o i n v es t i
g a t e r eac t i o n s an d t o ex p lo re m o lecu l a r g eo m et r i es .
Often th e dynam ics i s used as an efficien t me ans to se arch
fo r m in im a o f en erg y r a th er t h an t o r ev ea l t im e-
dependent effects .
Mo lec ular dy nam ics on f i tted surfac es i s also used to
co m p u te r eac t i o n r a t es u s in g c l as s i ca l s t a t i s t i ca l m eth o d s .
Somet imes the t rajector ies are used in a semiclassical for
m u l a t i o n t o o b t a i n q u a n t u m - m e c h a n i c a l a p p r o x i m a t i o n s .
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923
Th e so -ca l l ed e ik o n a l ap p ro x im at io n (Mo t t an d Massey ,
1965) der ives class ical equat ions for nuclear posi t ions and
m o m e n t a .
On e sem ic l ass ica l sch em e i s t h e wav e-p a ck e t d y n am ics
of He l ler (H ube r and Hel le r , 1987) . I t i s der ive d v ia the
t im e-d ep en d en t v a r i a t i o n a l p r in c ip l e (TDVP) b y co n s id
e r in g a ll t r an s l a t ed an d Ga l i l e i -b o o s t ed Gau ss i a n wav e
p ack e t s fo r t h e n u c l ea r co o rd in a t es as t h e m an i fo ld o f a l
l o wed wav e fu n c t io n s . Req u i r i n g t h a t t h e ac t i o n b e s t a
t i o n ary t h en y i e ld s t h e f am i l i a r Eu le r -Lag ran g e eq u a
t i o n s . Fo r t r an s l a t ed an d Gal i l e i -b o o s t ed Ga u ss i an wa v e
p ack e t s , t h ese co r resp o n d to c l as s i ca l Ham i l to n i an eq u a
t ions for the var iab les
R
a n d
P,
wh ich a re t h e av erag e n u
c l ea r p o s i t i o n s an d m o m en t a i n t h e Gau s s i an wav e p ac k
e t s . Th e m eth o d can b e easi l y ex t en d ed t o i n c lu d e a sca l
i n g p aram ete r , r e su l t i n g i n so -ca l l ed t h awed , as o p p o sed
to f ro zen , Gau ss i a n s . Th e ap p l i ca t i o n s wi th t h i s ad d i
t i o n a l d eg ree o f f r eed o m req u i r e ca re fu l i n t e rp re t a t i o n , a s
th e i r r esu l t s a r e so m et im es m i s l ead in g (Re im ers an d
Hel ler , 1988) .
T h e w a v e - p a c k e t d y n a m i c s m e t h o d h a s b e e n e x t e n d e d ,
by en larg ing the set o f al lowed wave funct ions in the
T D V P , t o d e s c r i b e m o r e c o m p l e x w a v e p a c k e t s c o n
s t ru c t ed as su p erp o s i t i o n s o f t im e-d ep en d en t b as i s fu n c
t i o n s (Ku car an d Mey er , 1 9 8 9 ) .
Wi th t im e co r re l a t i o n fu n c t i o n s t h ese m eth o d s can
y ie ld t r an s i t i o n p ro b ab i l i t i e s (Vi l l a lo n g a an d Mich a ,
1992).
b. Computed surface
A n in creas in g n u m b er o f wo rk er s n eed m o re access ib l e
su r faces t h en t h o se f i t t ed f ro m e l ec t ro n i c s t ru c tu re ca l cu
l a t i o n s i n o rd er t o s t u d y i n t e rm ed ia t e - s i ze sy s t em s .
Mo lecu l a r d y n am ics u s in g g rad i en t s f ro m a sem iem p i r i -
ca l Ham i l to n i an h as b een im p lem en ted b y S t ewar t i n
M O P A C ( S t e w a rt , 1 9 90 ), b y D e w a r i n A M P A C ( D e w a r
et aL,
1 9 85 ) , an d b y W ein er (Ca rm er
et aL,
1 9 9 3 ; Zh ao
et aL,
1993) and inde pen den t ly by Ed w ard s (1992) in
Z I N D O ( Z e r n e r , 1 9 91 ). T h e s e r e s e a r c h e r s h a v e i m p l e
m e n t e d N e w t o n i a n m o l e c u l a r - d y n a m i c s m e t h o d s u s i n g
d i r ec t l y ca l cu l a t ed g rad i en t s i n sem iem p i r i ca l e l ec t ro n i c
s t ru c tu re co d es . Th i s a l l o ws e ff ic ien t t im e-d ep e n d en t
s tud ie s of dyn am ics on a s ing le surfac e. Th e forces for a
s in g l e -d e t e rm in an t a l wav e fu n c t io n a re g iv en i n t h e Ap
pendix by Eq. (A86) .
4. Dynam ics of electrons and nuclei
So m e t im e-d ep en d en t m eth o d s i n c lu d e ex p l i c i t e l ec
t ro n i c d y n am ics as o p p o sed t o an av erag ed e l ec t ro n i c
d escr ip t i o n . Ex am p les i n c lu d e (i) d y n a m ics w i th
d en s i t y - fu n c t io n a l t h eo ry fo r e l ec t ro n s an d n u c l e i i n co n
d en sed p h ases as p ro p o sed b y Car an d Par r i n e l l o (1 9 8 5 ) ;
( ii) t h e tim e-d ep en d en t H ar t r ee - Fo c k
( T D H F )
m e t h o d
an d v ar i a t i o n a l ex t en s io n s b y Gazd y an d Mich a (1 9 8 6 ) ;
( ii i) T D H F fo r e l ec t ro n s wi th c l as s i ca l n u c l ea r d y n a m ics
o n an av erag e p o t en t i a l , d ev e lo p ed b y Mich a an d co
w o r k e r s ( R u n g e
et aL,
1 9 9 0; M ich a an d R u n g e , 1 9 9 2 ;
Ru ng e, 1993) for
ab initio
Ha m i l to n i an s , b y F i e ld (1 9 92 )
fo r sem iem p i r i ca l m eth o d s , an d b y Mik k e l sen an d Ra tn er
(1989) for elect ron t r ansfe r in so lvents ; ( iv ) dyn am ics on
an av erag e su r f ace o b t a in ed f ro m sev era l ad i ab a t i c e l ec
t ro n i c su r f aces (M ey er an d M i l l e r , 1 9 7 9 ; Ol so n an d
M ich a , 1 9 8 4 ) ; an d (v) c lo se -co u p l in g m e th o d s fo r a to m ic
col l i s ions wi th one (Fr i t s ch an d Lin , 1991) or two elec
t ro n s t r ea t ed ex p l i c i t ly (Kim u ra an d Lan e , 1 9 9 0) i n t h e
f ield of atomic co l l i s ions.
a. Wave-packet propagation on coupled potential surfaces
So m e m eth o d s g o b ey o n d th e r es t r i c t i o n o f n u c l ea r d y
n am ics o n a PES b y i n c lu d in g o n e o r m o re ex c i t ed e l ec
t ro n i c su r faces . Th e sam e m eth o d s fo r ex ac t p ro p ag a t io n
o f wav e p ack e t s o n a s i n g l e su r f ace h av e b een im p lem en t
ed fo r two co u p led su r faces . Th e Heid e lb erg g ro u p h as
made s ign i f ican t contr ibu t ions in th is f ield (Manthe and
K o p p e l , 1 9 9 0 b; M a n t h e
et aL,
1 9 9 1 ; Ko p p e l an d
M an t h e , 19 9 2 ). A fo u r th -o rd er -d if f e ren ce sch em e h as
b een im p le m en ted (Ma n th e an d Ko p p e l , 1 9 90 a) an d h as
been found to be more eff icien t than a second-order
difference by a factor of 3, bu t less efficient th an th e
sh o r t - t im e i t e r a t i v e Lan c zo s fo rm u la (K o p p e l an d
M an th e , 1 9 92 ) . Mu l t i co n f ig u ra t i o n a l t im e-d e p en d en t
H a r t r e e m e t h o d s w i t h n u m e r i c a l w a v e - p a c k e t p r o p a g a
t i o n o n co u p led su r faces h av e a l so b een im p le m en ted b y
M e y e r , C e d e r b a u m , a n d c o - w o r k e r s ( K u c a r
et aL,
1987;
M e y e r
et aL,
1 9 90 ; M a n t h e
et aL,
1992a, 1992b) . Th ese
f o r m u l a t i o n s a r e d e r i v e d f r o m t h e T D V P .
A d if f e ren t ap p ro ach t o n u m er i ca l wav e -p ack e t p ro p a
g a t i o n o n sev era l su r f aces i s t h e m eth o d o f t h e wav e-
p ack e t p e r tu rb a t i o n t h eo ry of Co a l so n an d co -w o rk er s ,
explain ed in detai l in th e rev iew of Co also n (1989). Th is
m e th o d i s we l l su i t ed for o p t i ca l sp ec t ro sco p y in a rea s
wh e re t h e s i n g l e -su r face ap p ro ac h es b reak d o wn . In t h e
fu l l fo rm u la t i o n , i t t r ea t s t h e wav e p ack e t s o n t h e
d i ff e ren t su r f aces as i n d ep en d e n t en t i ti e s . Pe r tu rb a t i o n
th eo ry i s u sed t o t r ea t t h e t r an s fe r o f p ro b ab i l i t y i n sp ace
an d t im e b e tween su r faces . Th i s m eth o d i s n o t v a r i a t i o n
al.
b. Trajectories on coupled potential surfaces
Th e m eth o d p ro p o sed b y Mey er an d Mi l l e r (1 9 7 9 ) p ro
v id es a g en era l f r am ewo rk i n wh ich t o em p lo y m u l t i p l e
elec t ron ic surfaces . I t cons iders on ly one set o f nuc lea r
co o rd in a t es m o v in g o n t h e av erag e p o t en t i a l an d ac t i o n -
an g le v a r i ab l es fo r e l ec t ro n i c d eg rees o f f r eed o m . Ol so n
an d M ich a (1 9 8 4 ) , u s in g t h i s ap p ro a ch , em p lo y t h e r ea l
an d im ag in ary p ar t s o f t h e e l ec t ro n i c am p l i t u d es as v a r i
ab les .
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c. Car-Parrinello method
T h e m e t h o d o f " p a r a l l e l d y n a m i c s " p r o p o s e d b y C a r
and Parr inel lo (1985) was f i rs t developed to make s imu
l a t ed an n ea l i n g m o r e e ff ic i en t. La t e r i t b eca m e q u i t e p o p
u l a r fo r d y n am ics s im u la t i o n s . Th i s m eth o d was r e
v i ewed r ecen t ly b y Rem ler an d Mad d en (1 9 9 0 ) an d b y
P a y n e s
al
(1992).
Car and Parr inel lo add f ict i t ious k inet ic-energy terms
to o b t a in d y n am ica l eq u a t io n s fo r t h e v a r i ab l es d esc r ib
in g t h e e l ec t ro n i c s t a t e . Th i s m eth o d h as cau sed co n s id
e rab l e d eb a t e , an d t h e re ap p ear s t o b e n o c l ea r u n d er
s tan ding of wh y it o f ten wo rks . I t i s def ined by co nside r
i n g t h e b as i c p a ram ete r s d esc r ib in g t h e e l ec t ro n i c s t a t e o f
th e sys tem t o be the o rbi tals t />; or the ir coefficients in
so m e ap p ro p r i a t e b as i s . Car an d Par r i n e l l o ap p ly t h e
m eth o d u s in g t h e d en s i t y - fu n c t io n a l d esc r ip t i o n o f ex
t en d ed sy s t em s , b u t , fo l lo win g Rem ler an d Ma d d e n in
th e i r ex p o s i t i o n o f t h e m eth o d , we ex p l a in t h e ap p ro ach
in g en era l t e rm s . A Lag ran g ian fo r t h e sy s t em o f t h e
elect ronic s tate and the nuclei i s def ined as
L = r - F + 2 A
i y
( ( ^ | ^ ) - 8
/ y
) (1.4)
U
wi th Lag ran g e m u l t i p l i e r s t o en su re o r th o n o rm al i t y o f
th e o rb i t a l s t h ro u g h o u t t h e d y n am ics . Th e k in e t i c en er
gy
L
i
z
k
has the usual terms for the nuclei , bu t i t a lso has terms
for the elect ro n ic pa ra me ter s . Th is energ y is cal led f icti
t i o u s b y Ca r an d Pa r r i n e l l o an d i s a p u re ly t ech n i ca l d e
v i ce t o d e r iv e d y n am ica l eq u a t io n s fo r t h e e l ec t ro n i c p a
r am ete r s f rom th e ab o v e Lag ran g ian . A sy s t em o f co u
p l ed eq u a t io n s fo r t h e n u c l e i is o b t a in ed , wh ich a re t h e
f a m i l i a r m o l e c u l a r - d y n a m i c s e q u a t i o n s . T h e a p p r o a c h
a l so g iv es a se t of eq u a t io n s fo r t h e e l ec t ro n i c p a ra m e te r s
wh ich p erm i t t h e p ro p ag a t io n o f t h e e l ec t ro n i c s t a t e " i n
p ara l l e l " wi th t h e n u c l ea r m o t io n . I t t u rn s o u t t h a t t h i s
is mo re efficien t tha n t ry in g to find the new op t im al elec
t ro n i c p a ram ete r s a t each g eo m et ry ; t h i s ap p ro ach a l so
y i e ld s e l ec t ro n i c s t a t es t h a t a r e v e ry c lo se t o t h e o p t im al
s tate s at a l l geom etr ies along the t rajec tory . In Sec.
I . B A g , t h e r e q u i r e m e n t s o n p a r a m e t r i z a t i o n f o r t i m e -
d ep e n d en t m eth o d s a re ex p l a in ed in d e t a i l . Becau se C ar
an d Par r i n e l l o d e f in e t h e i r p a ram ete r s fo r t h e e l ec t ro n i c
state to be real , the on ly way to ob tain equat ions i s by
ad d in g t h e " f i c t i t i o u s" en erg y wh ich i n t ro d u ces t h e
d e p e n d e n c e o n
ip .
Th e p ro p er ch o i ce i s t o u se co m p lex
p ar am ete r s . Th en t h e im ag in ary p ar t i s t h e co n ju g a t e
v ar i ab l e , t h e q u an tu m -m e ch a n ica l e l ec t ro n i c en erg y co n
t a i n s t h e p r o p e r d e p e n d e n c e o n c o o r d i n a t e s a n d m o m e n
t a , an d t h e Lag ran g ian f ro m th e t im e-d ep en d en t v a r i a
t i o n a l p r in c ip l e g iv es co r rec t eq u a t io n s wi th o u t ad d in g
t e r m s .
Th e adv an tag e of in t ro duc ing the f ict it ious
k in e t i c - en erg y t e rm s is t h a t o n e can g iv e t h e e l ec t ro n i c
v ar i ab l es a m ass co m p arab l e t o t h a t o f t h e n u c l e i , o r ev en
l a rg er . Th e r esu l ti n g d y n am ic s wil l t h en h av e t h e sam e
t im e sca l e as t h e n u c l ea r m o t io n , wh ich m ak es t h e p ro
ced ure very efficien t . A s a resu l t , howe ver , the dyn am ics
o f t h e e l ec t ro n i c v a r i ab l es b ear s n o r e l a t i o n sh ip t o an y
a p p r o x i m a t i o n o f t h e u n d e r ly i n g q u a n t u m - m e c h a n i c a l
d y n am ics o f t h e e l ec t ro n s .
Har tk e an d Car t e r (1 9 9 2 ) d ev e lo p an a l t e rn a t i v e way to
s im u la t e d y n am ics o n a PES a lo n g th e sam e g en era l l i n es
an d ap p ly i t t o a to m ic c lu s t e r s .
d. Time-dependent Hartree-Fock
Th ere i s a l a rg e c l as s o f m eth o d s t h a t a r e v a r i an t s o f
T D H F fo r t h e d y n am ic s of e l ec t ro n s an d t h a t em p lo y a
sem ic l ass i ca l o r c l as s i ca l d esc r ip t i o n fo r t h e a to m ic n u
c l e i . Ex am p le s a re t h e wo rk o f Ku lan d er an d co l l ab o ra
t o r s ( K u l a n d e r
et al.,
1 9 8 2 ; T i szau er an d K u lan d er ,
1 9 8 4, 1 9 9 1) ; o f Mich a , F en g , an d Ru n g e (Ru n g e
et al.,
1 9 9 0; Mi ch a an d R u n g e , 1 9 9 2 ; Ru n g e , 1 9 93) ; o f F i e ld
(1992); and of M ikkels en and R at ne r (1989). Al l these
m eth o d s co n s id er an ex p l i c i t d y n am ica l d esc r ip t i o n o f
th e e l ec t ro n i c s t a t e . So m et im es t h e fu l l
ab initio
H a m i l -
t o n i an i s co n s id ered (R u n g e an d Mic h a) ; so m e t im es a
m o d e l Ham i l to n i an i s se t u p t o d r iv e t h e d y n am ics (F i e ld ,
Mik k e l sen , an d Ra tn er ) . Th e co u p l in g b e tween th e e l ec
t ro n s a n d t h e n u c l e i i n t h ese m o d e l s i s t h ro u g h th e ( av er
age) po te n t ial -e nerg y surface. Th e nuclei feel the s urface
an d th e e l ec t ro n s f ee l t h e n u c l e i o n ly t h ro u g h th e i r i n
s t an t a n eo u s p o s i ti o n s i n t h e Fo ck o p e ra to r . As a resu l t ,
e l ec t ro n m o m en ta a re n o t t r ea t ed co r rec t l y , a d e f i c i en cy
wh ich sh o ws u p m ain ly i n h ig h er , n o n ch em ica l en erg y
reg im es . To r em ed y th i s p ro b l em , e l ec t ro n t r an s l a t i o n
fac to r s (ETF ' s ) a r e so m et im es i n t ro d u ced (Ba t es an d
Mc Ca r ro l l , 1 9 5 8 ; De lo s , 1 9 8 1 ; F r i t sc h an d L in , 1 9 9 1 ;
Riera, 1992) .
F i e ld h as ap p l ied TD H F a t t h e sem iem p i r i ca l Au s t i n
Mo d el 1 (AM I) l ev e l fo r s im u la t i o n o f t h e d y n am ics o f
L i H , H
2
0 , a n d C H
2
0 molecules (Field , 1992) . A br ief
d ef ini t io n o f t h e sem iem p i r i ca l Ha m i l to n i an u sed b y
Field i s p re sen t ed in Sec. 3 of the Ap pe ndi x . On e of the
m ain co n c lu s io n s o f t h ese s im u la t i o n s is t h a t t h e T D H F -
A M 1 m et h o d g iv es t r a j ec to r i es t h a t d i sp l ay r esu l t s for
v ar io u s s t a t i c an d d y n am ica l p ro p er t i e s (wh en a self-
consis ten t - f ield wav e funct ion i s chos en as a s ta r t ing
p o in t ) t h a t a r e eq u iv a l en t t o t h o se ca l cu l a t ed v i a t r a j ec
to r i es o b t a in ed wi th a m e th o d en su r in g t h a t t h e e l ec t ro n
i c v a r i ab l es sa t i s fy t h e t im e- in d ep en d en t v a r i a t i o n a l p r in
cip le at each t ime s tep . F ield claim s, base d on th is resu l t ,
t h a t t h e T D H F a p p r o a c h h a s fe w a d v a n t a g e s fo r t h e d y
namics of closed-shel l systems wi th wave funct ions ly ing
o n o r v e ry c lo se t o t h e Bo rn -O p p en h e im er su r f ace . Th i s
claim is addressed again in the appl icat ions sect ion (Sec.
IV) af ter i t i s shown in Sec. I I I .A.3 that F ield 's equat ions
o m i t t h e n o n ad iab a t i c co u p l in g t e rm s . He h as a l so i n co r
p o ra t ed t h e r ad i a t i o n f i e ld an d r ad i a t i o n -m at t e r i n t e rac
t i o n a t t h e lo n g -wav e l en g th d ip o l e ap p ro x im a t io n . As a
resu l t , i t i s possib le to s tudy the detai led dynamics of the
e l ec t ro n i c p o p u la t i o n i n sev era l s t a t es wh en a d o n o r -
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925
acc ep to r m o lecu l e i s ex c i t ed b y r ad i a t i o n o f v a r io u s
w a v e l e n g t h s .
K u l a n d e r a n d c o - w o r k e r s ( K u l a n d e r
et ah,
1982;
T i szau er an d Ku lan d er , 1 9 8 4 , 1 9 9 1 ) d esc r ib e t h e e l ec
t ron ic orb i tals numerical ly on a gr id f ixed in space and
p ro p ag a t e t h em to g e th er wi th t h e c l as s i ca l n u c l ea r p o s i
t i o n s as a co u p led sy s t em o f d i ff e ren ce eq u a t io n s . Th e i r
s tu d y i s r es t r i c t ed t o co l l i n ear r eac t i o n s i n t r i a to m ic sy s
t em s l i k e H
+
-f H
2
, fo r which the equat ion for the (dou
b ly o ccu p ied ) o rb i t a l 0 b eco m es
4d>(r,r) = A(D(r,f) ,
ot
h=-~^V
2
+V
e
(x,t)+V
ne
(x,t),
m
(1.6)
r- - -i- |r-4)e,| '
T h e e q u a t i o n s u s e d b y R u n g e a n d M i c h a ( M i c h a a n d
Ru n g e , 1 9 9 2 ; Ru n g e , 1 9 93) i n t ro d u c e an e leg an t so lu t i o n
to the problem of largely d i f fer ing t ime scales in the cou
p l ed sy s t em o f eq u a t io n s . Ru n g e an d Mich a s t a r t f ro m
t h e T D H F e q u a t i o n f or t h e d e n s i t y m a t r i x T,
it=Fr-rF, d.7)
w h e r e F i s t h e Fo c k m at r i x . Th e Fo c k m at r i x d e p en d s
l i n ear ly o n t h e d en s i t y m at r i x an d i s d esc r ib ed i n m o re
detai l in Eq . (3 .23) . The nuclei are t reated classical ly and
a l t e rn a t i v e ly fo l l o w p resc r ib ed t r a j ec to r i es , s t r a ig h t l i n es
o r Co u lo m b t r a j ec to r i es , o r t h ey fo l l o w t r a j ec to r i es co m
p u ted f ro m
M
k
K
k
=V
Kk
E( R
9
T)
, (1.8)
w h e r e t h e a v e r a g e p o t e n t i a l
E(R,T)
i s t h e ex p ec t a t i o n
v a lu e o f t h e m o lecu l a r Ham i l to n i an , i n c lu d in g t h e n u
c l ea r r ep u l s io n t e rm s an d t h e e l ec t ro n i c en erg y o f t h e
s t a t e d esc r ib ed b y t h e d en s i t y m at r i x T. To av o id h av in g
to i n t eg ra t e t h e f as t e l ec t ro n i c m o t io n i n Eq . (1 .7 ) , Mich a
an d Ru n g e l i n ear i ze t h e eq u a t io n d u r in g t im e s t ep s At,
l o n g co m p ar ed t o t h e e l ec t ro n i c t im e sca l e , b u t sh o r t fo r
th e n u c l e i , wi th t h e as su m p t io n t h a t t h e e f f ec t o f t h e n u
c l e i i s a sm al l p e r tu r b a t i o n o n t h e ev o lu t i o n o f t h e d en s i t y
m a t r i x . T h e y w r i t e r(t) = r(t) + r
l
(t), wh e re t h e r e fe r
en ce d en s i t y T
0
is p ro p ag a t ed as su m in g th a t t h e Fo ck
m at r ix r em ain s t h e sam e as a t t im e t
0
it
0
=F(t
0
)r
0
-rF(t
0
) (1.9)
w i t h F( t
Q
)=F(R(t
0
),r(t
0
)) . T h e c o r r e c t i o n T
1
t h en
gives the effect o f the mot ion of the nuclei , l inear in the
ch an g e o f t h e Fo ck m at r i x , o n t h e d en s i t y
it
l
=F(t
0
)r
l
-r
l
F(t
0
)+AFr-r&F , d.io)
w h e r e AF=F(R (t)
9
T(t))F(t
0
). Th es e eq u a t io n s a re
in t eg ra t ed f ro m t
0
to t
0
+ At b y d i ag o n a l i z in g F( t
0
) an d
w r i t i n g r a n d T
1
as a su p erp o s i t i o n o f t h e e ig en m o d es .
A n eff icien t algor i thm is used to incre ase or decrea se At
d u r in g t h e ev o lu t i o n as n eed e d . Ru n g e an d M ich a wr i t e
th e i r eq u a t io n s i n t h e t r av e l i n g a to m ic-o rb i t a l b as i s ,
wh ic h i s im p o r t a n t fo r t h e q u a l i ty o f t h e i r r esu l t s (Mich a
an d Ru n g e , 1 9 9 2 ; Ru n g e , 1 99 3) . A m o re r ecen t t r ea t
m e n t (Mich a , 1 9 94 ) u s in g L io u v i l le o p era to r s g e n era l i zes
th ese eq u a t io n s t o a l l o rd er s i n AF.
e. Time-dependent density functional
Rec en t ly Th e i lh ab e r (1 9 92 ) im p le m en ted t h e r i g o ro u s
t im e ev o lu t i o n u sed i n t h e T D H F in t h e fi eld o f d en s i t y -
fu n c t io n a l t h e o ry fo r ex t en d ed sy s t em s as an a l t e rn a t i v e
t o t h e C a r - P a r r i n e l l o m e t h o d . H e d e s c r i b e s t h e e l e c t r o n
i c sy s t em b y u s in g Ko h n -Sh am o rb i t a l s an d o b t a in s t h e
d y n a m i c a l e q u a t i o n s
r\ 1
ix[>j(r,t)=-V
2
xl>j(r,t) + v
ef[
(r,t,[n])il>j(T,t) ,
2
m
( L I D
M^R
k
(t)=F
k
(t) ,
w h e r e
F
k
(t)
i s the to ta l force on the fcth ion and w he re,
wi th d o u b le o ccu p an cy , t h e d en s i t y i s g iv en b y
N
e
/2
n ( r , f )=
2
2 - l ^
r
> * > l
2
(
L 1 2 )
7 = 1
These equat ions wi l l be s tud ied in detai l in Sec. I l l fo r
th e s i n g l e -d e t e rm in an t a l w av e fu n c t io n [ see Eq s .
( 3 .2 2 ) - (3 .2 5 ) ] an d a re d e r iv ed f ro m th e T D V P in Sec . 1 o f
th e Ap p en d ix . M o re d e t a i l s can b e fo u n d in Th e i lh ab er ' s
p ap er (Th e i lh ab er , 1 9 9 2 ) . He p o in t s o u t t h a t t h e t im e-
d e p e n d e n t d e n s i t y - f u n c t i o n a l ( T D D F ) a p p r o a c h h a s a
p h y s i ca l k in e t i c en e rg y as o p p o sed t o t h e f i c t it i o us k in e t i c
en erg y of Car an d Par r i n e l l o (1 98 5 ) . Fu r th erm o re , b e
cau se o f t h e r i g o ro u s r e l a t i o n t o t h e Sch ro d in g er eq u a
t i o n fo r t h e ful l sy s t em , t h e T D D F eq u a t io n s co n serv e t o
t a l m o m e n t u m a n d t o t a l e n e r g y .
To av o id h av in g t o i n t eg ra t e t h e f as t e l ec t ro n i c m o t io n
fo r l o n g t im es , Th e i lh ab er u ses a p ro ced u re wel l k n o wn
in the f ield of p la sm a physics (Birdsa l l an d La ng do n,
1 9 91 ) ; i . e ., t h e co m p u te r s im u la t i o n s a re r u n w i th t h e i o n
m asses eq u a l t o 1 0 m
e
, wh ic h i s sm al l en o u g h t o a l l o w fo r
p i co seco n d s im u la t i o n s , b u t s t i l l l a rg e en o u g h to en su re
ap p ro x im ate ad i ab a t i c i t y o f e l ec t ro n m o t io n i n r esp o n se
to io n d i sp l acem e n t s . Th e r esu l t s a r e t h en r esca l ed t o
m ass r a t i o s o f i n t e res t .
f. dose coupling and perturbed stationary state
Rev iews o f t h e sem ic l as s i ca l an d q u an tu m v er s io n s o f
th e c lo se -co u p l in g ap p ro ach a re g iv en b y Delo s (1 9 8 1 ) ,
K im u ra a n d Lan e (19 9 0 ), an d F r i t s ch an d L in (1 9 9 1) . I t
i s a m eth o d to d esc r ib e ch arg e- t r an s fe r p ro cesses i n s l o w
(Delos, 1981) and , recent ly , a lso fast (Riera, 1992) atomic
co l l i s i o n s . On e co n s id er s a t a rg e t sy s t em o r ig in a l l y a t
r es t a t t h e o r ig in an d a p ro j ec t i l e a to m ap p ro ach in g t h e
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t a rge t w i th g iven impa c t pa ram e te rs and ve loc i ty . T h e
me thod concen t ra te s on des c r ib ing one ac t ive e lec t ron ,
the others being frozen in core orbi ta ls or t rea ted by
ps eudop o ten t i a l s . Some s ys tems wi th two ac t ive e lec
t ron s hav e been s tudied (Fri tsch and Lin, 1991). Th e
semiclass ica l form of the met ho d is briefly d iscussed
below . Delo s (1981) gives a deta i led discuss ion of the ful
ly quan tum -mec han ic a l fo rm of the c los e -coup l ing
m e t h o d .
Th e m eth od has thre e ingredien ts : ( i) a choice of nu
c lear t ra jec tory, usual ly a prescribed t ra jec tory, often a
s tra ight l ine or Coulomb tra jec tory; ( i i ) a choice of bas is
se t for the e lec tro nic wave func t ion; a nd ( i ii ) the solut ion
of coupled different ia l equat ions in t ime for the
coeffic ients of the e lec tron ic wave funct ion . Th e choic e
of bas is se t in the c lose-coupl ing method has a r ich his to
ry reviewed in deta i l by Fri tsch an d Lin (1991). Th e
present consensus is to use molecular orbi ta ls xff
f
d e p e n d
ing on a l l nuclear coordinates
R
wi th e l ec t ron t rans la t ion
factors of the form (Kimura and Lane, 1990)
F ^ , r ) = e x p [ / ( m v T /
/
U , r ) - m u
2
^ / 2 ) / ^ ] ( 1 . 1 3 )
wi th a s wi tch ing func t ion / , wh ich has the a s ymp to t i c
values 1 for the l imit where the two a toms are far
apa r t . T he s pace - independen t k ine t i c -ene rgy t e rm can be
left off and incorpora ted in the wave-funct ion expans ion
coeffic ients . Th e ques t ion of ET F' s is addr essed in deta i l
in Sec. II.A.
The tota l molecular wave funct ion is then wri t ten as
V(R,r) = ^Xi(Rtyi(R,r)Fi(R>r) (1.14)
In the s emic la s s ica l approx ima t ion , th i s becomes
mR
y
r) = ^a
i
(t)if;
i
[R(tU]F
i
[R(tU]exp(f) , (1.15)
i
where
f = -i f
t
E
i
[R(t
,
)]dt
,
~^ f
t
v
2
dt
f
. (1.16)
Subs tituting E q. (1.15) in the time-de penden t Schrodin ger
equation, projecting on the electronic basis, and expand
ing to first order in v give the coupled equation s
iS^
L
= [h+v-(V+ A)]a
(1.17)
at
with
P,7 =
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927
i d en t i t y . M ath em at i ca l l y , co h e ren t s t a t es a r e an o v er -
co m p le t e se t wi th a r eso lu t i o n o f t h e i d en t i t y . Th e f ac t
t h a t a l l s t a t es i n t h e o v erco m p le t e se t ad d u p t o u n i ty
m a k e s t h e s e t " c o h e r e n t . " T h e u n d e r s t a n d i n g o f th e
th eo r y o f co h eren t s t a t es an d L ie g ro u p s i s i n n o w ay
essen t i a l t o t h e u n d er s t an d in g o r u se o f t h e t h eo ry , o r t o
th e d er iv a t i o n o f t h e p h y s i ca l p ro p er t i e s o f t h e eq u a t io n s .
Th e t h eo ry o f co h e ren t s t a t es is i n sp i r i n g an d im p o r t a n t
i n o rd e r t o fi nd t h e o p t im a l p a ram et r i z a t i o n o f t h e w av e
funct io n . I t i s a lso usefu l in prov ing the in te rna l con
s i s t en cy o f t h e eq u a t io n s . Co h eren t s t a t es an d L ie g ro u p s
are n o t i n t ro d u ced i n t h i s p resen t a t i o n o f t im e-d ep en d en t
t r ea tm en t s , b u t t h e i r r e l ev an ce i s c l a r i f i ed i n Sec . I I .B .2
fo r t h e i n t e res t ed r ead er .
Th e EN D th e o ry can b e co n s id ere d as an ex t en s io n of
th e TD H F m e th o d s i n t h e sense t h a t e l ec t ro n s an d n u c l e i
a re a l l o wed to i n t e rac t wi th o u t an y r es t r i c t i o n , a s o p
p o sed t o m e th o d s wi th a p o t en t i a l su rf ace co n s t ru c t i o n
an d th e as so c i a t ed av erag in g o v er e l ec t ro n i c m o t io n .
Th i s im p l i es , am o n g o th er t h in g s , t h a t n e i t h e r t h e Bo rn -
O p p e n h e i m e r n o r t h e a d i a b a t i c a p p r o x i m a t i o n i s e n
fo rced i n EN D an d th a t ( ev en wi th c l as s i ca l n u c l e i ) t h e
t r a j ec to r i es a r e t ru ly d y n am ica l an d r esu l t f ro m th e ac
t i o n o f t h e i n s t an t an eo u s fo rces .
E N D d if fe r s f ro m o th er t im e-d ep en d en t m eth o d s p ro
p o sed an d d ev e lo p ed i n r ecen t y ear s i n t h a t i t r eco g n izes
th a t t h e e l ec t ro n -n u c l ea r d y n am ics t ak es p l ace i n a g en
e ra l i zed p h ase sp ace . Th e d e t a i l ed an a ly s i s o f t h i s p h ase
sp ace an d t h e ex p l i c i t co n s t ru c t i o n o f i t s m et r i c a r e im
p o r t a n t i n g r e d i e n t s i n t h e E N D a p p r o a c h . O t h e r
m eth o d s as su m e, im p l i c i t l y , t h a t t h e p h ase sp ace i s
can onic al , i.e . , f lat . In some cases , tha t ma y be corr ect ;
i n o th e r s , t h o u g h , t h i s r eq u i r es fu r th e r i n v es t i g a t i o n . Fo r
i n s t a n c e , w h e n E T F ' s a r e i m p o r t a n t , t h e c o r r e c t f o r c e s
th a t h av e cau sed s ig n i f i can t d eb a t e an d t h a t can b e d e
r i v ed f ro m ETF ' s b y so m et im es l ab o r io u s sch em es
(Delo s , 1 9 8 1 ) a re ac tu a l l y co n t a in ed i n t h e p h ase- sp ace
m e t r i c . Th e fo rm u la t i o n o f t h e d y n a m ics i n t h e p ro p er
p h ase sp ace g rea t l y c l a r i f i e s t h e d y n am ica l o r ig in o f t h ese
t e rm s an d s im p l i f i e s t h e i r d e r iv a t i o n .
Th e p r in c ip l es of E N D are few an d s im p le .
(1 ) Param et r i ze a wav e fu n c t io n fo r t h e m o lecu l e as a
wh ole, i .e . , fo r elec t rons and nucle i . M ak e sure tha t (a)
t h e p a r a m e t e r s a r e n o n r e d u n d a n t ; (b ) t h e p a r a m e t e r s a r e
d iv id ed i n to co o rd in a t es an d t h e i r co n ju g a t e m o m en ta ;
an d ( c ) t h e p a r am ete r s g en era t e a co m p le t e se t of wav e
fu n c t io n s .
(2 ) M ak e a l l p a ra m e te r s t im e d ep en d en t an d d er iv e
d y n am ica l eq u a t io n s fo r t h em u s in g t h e ch o sen
p ara m e t r i ze d wav e fu n c t io n s as t h e fam i ly o f a l l o wed
v ar i a t i o n s i n t h e T D V P. Al t h o u g h sev era l t y p es o f wav e
fu n c t io n s h av e b een co n s id ered (Deu m en s
et al.,
1987a,
1991; D e u m e n s a n d O h r n , 1 9 89 b ; W e i n e r et al, 1991),
t h i s r ev i ew co n cen t r a t es fo r sev era l r easo n s o n t h e s im
p l es t p o ss ib l e ch o i ce , i . e . , a s i n g l e d e t e rm in an tsp in
u n res t r i c t ed an d wi th co m p lex co ef f i c i en t sfo r t h e e l ec
t ro n s an d a c l as s i ca l t r ea tm en t o f t h e n u c l e i .
Am o n g th e r easo n s fo r t h i s ch o i ce a re t h e fo l l o win g :
( i) I t i s gene ral eno ugh to ex hib i t mo st in t r ica cies of
t im e-d ep en d en t t r ea tm e n t s ex p l i c i t l y ; ( ii) t h e ch o i ce i s
suff icien t ly r ic h to al low a me aning fu l co mp aris on w i th
ex p er im en t s fo r a l a rg e v ar i e ty o f p h y s i ca l an d ch em ica l
p ro cesses ; ( ii i) t h e g en era l f r am e wo rk g en era t ed b y t h e
ch o ice i n c lu d es m an y o f t h e m o s t wid e ly u sed m eth o d s as
sp ec i a l cases ; an d ( iv ) n ecessa ry g en era l i za t i o n s c an b e
fo rm u la t ed as co n cep tu a l l y s im p le ex t en s io n s , b u t w o u ld
b e cu m b erso m e to d i scu ss fo r t h e p u rp o ses o f a r ev i ew.
T h e p a r a m e t e r s s h o u l d b e s u c h t h a t e a c h q u a n t u m -
m e ch an ica l s t a t e , i. e . , each wa v e fu n c t io n u p t o a n o rm al
i za t i o n co n s t a n t an d g lo b a l p h ase f ac to r , i s m a p p e d o n e-
t o - o n e o n a s e t o f p a r a m e t e r v a l u e s . F o r T D H F , o n e
of ten uses mo lec ular -orb i tal coeff icien ts . Be caus e of the
wel l -k n o wn in v ar i an ce o f a d e t e rm in an t a l s t a t e u n d er a r
b i t r a r y t r an s fo rm a t io n s o f t h e o ccu p ied ( an d u n o ccu p ied )
o r b i ta l s a m o n g t h e m s e l v e s , t h e r e a r e m a n y p a r a m e t e r
v a lu es t h a t r ep rese n t t h e sam e s t a t e . Becau se t h e
Sch ro d in g er eq u a t io n d e t e rm in es t h e ev o lu t i o n o f q u an
tu m s t a t es , i t i s i n v ar i a n t u n d e r an y t r an s fo rm at io n t h a t
l eav es s t a t es i n v ar i a n t . As a r esu l t , t h e eq u a t io n s fo r t h e
p ara m e te r s d e r iv ed fro m th e TD V P wi l l b e i n v ar i an t a s
wel l . I t f o ll o ws t h a t su ch eq u a t io n s wi ll n o t d e t e r m in e
th e ev o lu t i o n o f r ed u n d an t p a ra m e te r s . Th i s l ead s t o n u
m er i ca l i n s t ab i l i t i e s . Th ese can b e e l im in a t ed b y ap
p ro p r i a t e co n s t r a in t s . Ho we v er , it i s o b v io u s t h a t t h e
system w i th con st ra in ts wi l l be less efficien t th an a
s t r a i g h t f o r w a r d p r o p a g a t i o n o f a s e t o f n o n r e d u n d a n t p a
r am ete r s . Fo r a s i n g le d e t e rm i n an t , t h e t h eo ry o f
co h eren t s t a t es as so c i a t ed wi th t h e u n i t a ry g ro u p o f t h e
s in g l e -p ar t i c l e sp ace im m ed ia t e ly y i e ld s t h e co r rec t p a
r a m e t e r s ( K r a m e r a n d S a r a c e n o , 1 9 8 1 ) . T h e s e p a r a m e
t e r s a r e k n o wn in n u c l ea r p h y s i cs as t h e Th o u less r ep re
sen t a t i o n (Th o u less , 1 9 6 0 ) o f a d e t e rm in an t a l s t a t e an d
are used in the class i f icat ion of sp in- and charge-densi ty
wav es i n so l i d - s t a t e t h eo ry (F u k u to m e , 1 9 8 1) . T h e i r co n
s t ru c t i o n i s p resen t ed i n Sec . I I .B , an d i n Sec . I I I .C .2 i t is
p r o v e n t h a t t h e y c a n b e i n t e r p r e t e d a s r a n d o m - p h a s e -
a p p r o x i m a t i o n ( R P A ) a m p l i t u d e s ( L i n d e r b e r g a n d O h r n ,
1973).
Th e seco n d r eq u i r e m e n t o n t h e p aram ete r s is t h a t t h ey
b e su i t ab l e t o d esc r ib e a d y n a m ic a l sy s t em . Th e
Sch ro d in g er eq u a t io n i n Hi lb er t sp ace i s an i n f in i t e -
d im en s io n a l , l i n ea r d y n am ica l sy s t em , an d t h e co m p lex
n a t u re o f Hi lb e r t sp ace is e s sen t i a l . Fo r t h e t im e-
d ep en d en t t r ea tm en t t o r e t a in t h a t e s sen t i a l f ea tu re , i t i s
n ecessa ry an d su f f i c i en t t h a t t h e p a ram ete r s fo rm a p h ase
sp ace , wh ich m ean s t h ey can b e d iv id ed i n to a se t o f
co o rd in a t es an d a se t o f co n ju g a t e m o m e n ta . Su ch a
d iv i s io n i s ca l l ed a sy m p lec t i c s t ru c tu r e an d i s t h e fo u n
d a t i o n o f t h e t h eo ry o f d y n am ica l sy s t em s (Go ld s t e in ,
1 9 80 ) . Im p o s i t i o n o f su ch a s t ru c tu re en su res t h a t t h e
t i m e - d e p e n d e n t t r e a t m e n t , h o w e v e r a p p r o x i m a t e , w i l l b e
d y n am ica l l y r easo n ab le . Th e e l em e n t s o f t h e o n e-p ar t i c l e
d en s i t y m at r i x p ro v id e a ch o i ce o f p a r am ete r s ( co o rd i
n a t es ) fo r a s i n g l e -d e t e rm in an t a l wav e fu n c t io n (Ru n g e ,
1 9 93) . H o we v er , n o co n ju g a t e m o m e n ta ex i s t an d t h e r e
fo re so m e in co n s i s t en c i es can b e ex p ec t ed fo r su ch a
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ch o ice . Becau se Ru n g e an d Mich a u se a l i n ear i zed fo rm
o f t h e eq u a t io n s t o p ro p ag a t e t h e e l ec t ro n i c co o rd in a t es ,
t h ey im p l i c i t l y u se co n ju g a t e m o m en ta t h a t a r e , a s fo r
an y o sc i l l a to r ,
rr/2
o u t o f p h ase ; an d n o p ro b l em s a r i se .
Were o n e t o a t t em p t a so lu t i o n o f t h e n o n l in ear eq u a
t i o n s , o n e m u s t d e f in e co n ju g a t e m o m en ta fo r t h e e l ec
t ro n i c d eg rees o f f r eed o m . W h en th e t h eo ry o f co h e ren t
s t a t es an d L ie g ro u p s i s u sed , t h e ex i s t en ce o f t h e sy m -
p l e c t ic s t r u c t u r e i s g u a r a n t e e d a n o t h e r a d v a n t a g e o f
u s in g t h ese m ath em at i ca l t o o l s .
Th e t h i rd r eq u i r e m e n t is t h a t t h e p a ra m e te r s b e su ch
th a t t h e wav e fu n c t io n s fo r a l l p o ss ib l e p a ram ete r v a lu es ,
i n p r in c ip l e , fo rm a co m p le t e se t . I t is fo rm al ly o v erco m -
p le t e b ecau se o f t h e co n t in u i ty o f t h e p a r am ete r s .
In o u r case t h i s m ean s t h a t t h e s i n g l e -d e t e rm in an t a l
wave funct ion for the elect rons i s
d e t ( X / , ( r J ) ( 1 . 1 9 )
wi th
X
h
(T)
= i>
h
(T) +
^^
p
(T)
Zph
,
(1.20)
P
w h e re t he ^ ( r ) , k=p,h, are a p p r o p r i a t e o r t h o n o r m a l
two -co m p o n en t sp in -o rb i t a l s ex p ressed i n so m e ( i n p r in
c ip l e , co m p le t e ) b as i s o f sp in -o rb i t a l s . Th i s p a ra m e t r i za -
t i o n en su res t h a t d u r in g t h e t im e ev o lu t i o n o f t h e z p a
rameters the system wi l l be ab le to access al l possib le
d e t e rm in an t a l wav e fu n c t io n s i n t h e g iv en o rb i t a l b as i s .
Th eo re t i ca l l y t h e m o s t co n v en ien t o rb i t a l b as i s wo u ld b e
a fi xed o r th o n o rm al se t . Th i s l ead s t o s im p le d y n am ica l
eq u a t io n s . On e ch o i ce wo u ld b e h arm o n ic -o sc i l l a to r
e ig en fu n c t io n s cen t e red a t t h e o r ig in . Th ey a re co m p le t e ,
o r th o n o rm al , an d easy t o wo rk wi th . On th e o th e r h an d ,
f ro m th e p o in t o f v i ew o f co m p u ta t i o n s , a d esc r ip t i o n i s
d es i r ed wi th p h y s i ca l q u an t i t i e s r ep resen t ed accu ra t e ly
b y a sm al l n u m b er o f t e rm s . S in ce r ep res en t a t i o n s o f o r-
b i t a l s o n o n e cen t e r i n a b as i s l o ca t ed a t an o th er cen t e r
co n v erg e p o o r ly , e l ec t ro n i c s t ru c tu re t h eo ry wo rk s wi th
a to m ic o rb i t a l s o n a l l cen t e r s . Th e sam e id ea wo rk s fo r
d y n am ica l p ro b l em s wi th ap p ro p r i a t e ad ju s tm en t s o f t h e
d y n am ica l eq u a t io n s . E l ec t ro n t r an s l a t i o n f ac to r s a r e a
s im i l a r co n v erg en ce acce l e ra t i n g d ev i ce . Wh en u sed wi th
th e co r rec t d y n am ica l eq u a t io n s , t h ey a re n o th in g m o re .
Deta i l s a r e d i scu ssed i n Sec . I I .A .
With a g iven choice of the form of the wave funct ion
an d wi th a ch o i ce o f p a ram ete r s ( i n c lu d in g o rb i t a l b as i s ) ,
a l l ap p ro x im a t io n s a re sp ec if i ed . Th e d er iv a t i o n an d
so lu t i o n o f t h e eq u a t io n s i n v o lv e n o fu r th e r ap p r o x im a
t i o n s .
II .
PREPARATIONS
Al l t r ea tm en t s o f m o lecu l a r sy s t em s u se q u an tu m
m ec h an ic s t o d esc r ib e t h e e l ec t ro n s . So m e m eth o d s o n ly
lo o k a t t h e e l ec t ro n s t h ro u g h th e e l ec t ro n i c e ig en s t a t es
asso c i a t ed wi th t h e PES ' s ; o th e r s t r ea t t h e e l ec t ro n i c d e
g rees o f f r eed o m d y n am ica l l y . Al th o u g h th e m eth o d s
that curren t ly use f i t ted surfaces have in tegrated out al l
d e t a i l ed e l ec t ro n i c i n fo rm at io n , t h e m o re ex ac t o n es , l i k e
t h e p s e u d o s p e c t r a l F o u r i e r a p p r o x i m a t i o n , a r e e x p e c t e d
to i n c lu d e ev en tu a l l y an ex p l i c i t t r ea tm en t o f t h e e l ec
t ron s as a wa y to ove rco me the l im i tat ion s of the f it ting
p r o c e d u r e . F u r t h e r m o r e , a ll m e t h o d s t h a t d o i n c l u d e a
fu l l t r ea tm en t o f t h e e l ec t ro n s n eed t o b e su p p l i ed wi th
an in i t ia l s ta te for the evo lu t ion . Th is is o f ten a mole cu
l a r s t a t e i n c lu d in g an e l ec t ro n i c s t a t e o n t h e PE S . A
th o ro u g h u n d er s t an d in g o f h o w th e e l ec t ro n s a re d e
sc r ib ed is e s sen t i a l fo r a l l t im e-d ep en d en t m e th o d s . Th i s
is the subject o f Sees . I I .A and I I .B.
Next , the descr ip t ion of the nuclei i s d iscussed in Sec.
I I .C, and f inal ly , in Sec. I I .D, a detai led d iscussion of the
t im e-d ep en d en t v a r i a t i o n a l p r in c ip l e (TDVP) i s g iv en .
A. Electronic spin-orbitals
1.
Choice of basis and convergence
An ap p ro x im ate m an y -e l ec t ro n wav e fu n c t io n can b e
rep rese nted in a var ie ty of wa ys. Bo th for the ease of in
t e rp re t a t i o n i n t e rm s of ch em ica l an d p h y s i ca l co n cep t s
and for computat ional eff iciency , i t i s convenien t to bu i ld
m an y -e l ec t ro n wav e fu n c t io n s f ro m s in g l e -p ar t i c le fu n c
t i o n s o r sp in -o rb i t a l s . Th es e o rb i t a l s a r e , i n g en era l ,
chosen to be expressed in terms of a basis set o f funct ions
of som e ana ly t ic form tha t can be eff icient ly ma nip ula ted .
In p r in c ip l e , o n e co u ld r ep resen t t h e m an y -e l ec t ro n fu n c
t i o n s o n a g r id an d o b t a in an "ex ac t " n u m er i ca l r ep resen
t a t i o n . Ho w ev er , t h e n u m b er o f g r id p o in t s i n c reases so
rap id ly wi th t h e n u m b er o f e l ec t ro n s , t h a t t h i s t ech n iq u e
has never been appl ied successfu l ly to more than one or
two e l ec t ro n s .
Th e choice of basis set in term s of wh ich to r epre sen t
th e sp in -o rb i t a l s h as b een g iv en m u ch co n s id era t i o n i n
q u a n tu m ch em is t ry . Ex c lu d in g e l ec t ro n sca t t e r i n g o r
io n i za t i o n p ro ces ses , i t i s c l ea r t h a t e l ec t ro n i c d en s i t y
rem ain s n ear t h e a to m ic n u c l e i t h ro u g h o u t a p ro cess . I f
o n e co n s id er s t h e n u m b er o f b as i s fu n c t i o n s o f so m e ty p e ,
say , h a rm o n ic -o sc i l l a to r - t y p e fu n c t i o n s cen t e red a t t h e
o r ig in , n eed ed t o r ep resen t an o rb i t a l l o ca t ed o n a n u
c l eu s so m e d i s t an ce r em o v ed , i t b eco m es c l ea r t h a t i t i s
more eff icien t to analy t ical ly move the basis funct ions
o v er t o t h e a to m ic n u c l eu s . W h en o n e cen t e r s b as is fu n c
t ions on a number of d i f feren t nuclei in that way , the re
su l t i n g t o t a l b as i s i s n o l o n g er o r th o n o rm al an d can i n
t r o d u c e a n n o y i n g a p p r o x i m a t e l i n e a r d e p e n d e n c i e s ; h o w
ever , i t i s known in pract ice to work qui te wel l for gen
e ra l m o le cu l a r sy s t em s . Th u s u s in g b as i s fu n c t i o n s cen
tered on the atomic nuclei rather than on f ixed poin ts in
sp ace i s an o b v io u s ch o i ce t h a t l ead s t o b e t t e r co n v er
g en ce p ro p er t i e s .
A s im i l a r o b serv a t i o n h o ld s wi th r eg ard t o co n v er
g en ce o f a b as i s fo r t h e v e lo c i t y o r m o m en tu m p ro p er t i e s
o f t h e e l ec t ro n s . Giv e n t h e p ro p er ly d e r iv ed eq u a t io n s o f
m o t io n , co n v erg en ce i s acce l e ra t ed b y an a ly t i ca l l y m o v -
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i n g t h e b as i s fu n c t i o n t o t h e a to m ic n u c l e i i n p h a se sp a ce ,
r a th e r t h a n j u s t i n co n f ig u ra t i o n sp ac e . Th i s s im p li f ie s
an d ex p l a in s t h e ro l e o f e l ec t ro n t r an s l a t i o n f ac to r s , a
su b j ec t t h a t h as r ece iv ed co n s id erab l e a t t en t i o n i n t h e
t im e-d ep en d en t fo rm u la t i o n s (De lo s , 1 9 8 1 ) .
2.
Atomic spin-orbitals
Th e elect ron ic wave funct ion i s u l t im ately def ined in
terms of a set o f
K
l o ca l ize
