Supercomputers in Chemistry - American Chemical Society

3

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on April 3, 2018 | https://pubs.acs.org Publication Date: November 6, 1981 | doi: 10.1021/bk-1981-0173.ch003

Supercomputer Requirements for Theoretical Chemistry ROBERT B. WALKER, P. JEFFREY HAY, and HAROLD W. GALBRAITH Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545

The electronic computer is undeniably an essential component in the tool bag of the modern theoretical chemist and, with ever improved accessibilty to more powerful computing, it is tempting to feel a sense of euphoria about current computer capabilities. At Los Alamos, we have a truly impressive resource of large computers -- at present we have a choice of two 1-million word CRAY machines and four CDC 7600 machines. However, even with this powerful a computing environment, the main objective of this talk is to ask whether or not this current euphoria is really justified. Have we yet reached the stage where we can, with current computers, address problems which are traditionally thought of as chemistry? The answer is, in many cases, no. In this talk, we survey three areas of theoretical chemistry which receive considerable attention at the Theoretical Chemistry and Molecular Physics Group at Los Alamos. These three areas are (1) molecular electronic structure calculations, (2) chemical dynamics calculations, and (3) quantum optics and spectroscopy. In introducing each category, we will note the types of mathematical algorithms used to solve problems typical of each area. We then present examples of types of calculations which we feel are at the current state-of-the-art. Finally, we will present a wish list of problems in each category which we would like to be able to study, but are simply beyond current computing capabilities. The program of this symposium makes it clear that there will be several talks to follow which will concentrate specifically on problems associated with electronic structure calculations — we will only skim over the subject for now. We will concentrate more heavily on problems in chemical dynamics, and conclude with problems in quantum optics.

0097-6156/81/0173-0047$05.00/0 © 1981 American Chemical Society

Lykos and Shavitt; Supercomputers in Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

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Molecular E l e c t r o n i c Structure Calculations In determining the quantum mechanical s t r u c t u r e of a molecule, there are three major steps (roughly equal i n d i f f i c u l t y ) to be attacked computationally (See F i g . 1.). We f i r s t d e f i n e a set of atomic b a s i s f u n c t i o n s centered at each nucleus and then compute a l a r g e number of i n t e g r a l s over these b a s i s f u n c t i o n s . This information feeds i n t o the c o n s t r u c t i o n of the Fock matrix. The eigenvalues of the Fock matrix are a s s o c i a t e d with the t o t a l energy of the system. The eigenvectors d e f i n e the occupied o r b i t a l s of the system and the eigenvalues d e f i n e the o r b i t a l energies. This information i s used to c o n s t r u c t a new Fock matrix, which i s again d i a g o n a l i z e d . This procedure i s repeated i n t e r a t i v e l y u n t i l the t o t a l energy of the system i s minimized and the o r b i t a l s are constant from one i t e r a t i o n to the next. Reasonable determinations of molecular s t r u c t u r e can be obtained at t h i s S C F - l e v e l of c a l c u l a t i o n . However, f o r an accurate determination of the s t r u c t u r e and p r o p e r t i e s of molecules, c o r r e l a t i o n s between the motions of the many e l e c t r o n s of the system must be i n c l u d e d . To do t h i s , many-electron wavefunctions are computed u s i n g sums of products of these one-electron o r b i t a l s . This process i s the c o n f i g u r a t i o n i n t e r a c t i o n (CI) method; g e t t i n g accurate CI wavefunctions and energies r e q u i r e s an enormously l a r g e b a s i s of SCF f u n c t i o n s . The Hamiltonian matrix i n t h i s b a s i s i s constructed and d i a g o n a l i z e d to get the accurate CI wavefunctions and energies. The CI matrix tends to be both very l a r g e and very sparse. E f f i c i e n t computer codes must take i n t o account the s p a r s i t y of the CI matrix both i n i t s c o n s t r u c t i o n and d i a g o n a l i z a t i o n phase. L e t ' s turn our a t t e n t i o n to the computational needs of the s t r u c t u r e problem (See F i g . 2). D e f i n i n g n as the number of atomic b a s i s f u n c t i o n s employed, there are s e v e r a l c h a r a c t e r i s t i c matrices to consider. The Fock matrix, which we repeatedly c o n s t r u c t and d i a g o n a l i z e u n t i l convergence, i s only an n x n matrix; u n f o r t u n a t e l y , to construct t h i s matrijc at each i t e r a t i o n of the SCF procedure, we have to process the n /8 two-electron i n t e g r a l s . At the current s t a t e of the a r t , the number of atomic b a s i s f u n c t i o n s n tends to be about 100. This l i m i t a t i o n i s not so much because of the d i f f i c u l t y of d i a g o n a l i z i n g 100 x 100 m a t r i c e s , but because of the 10 l i m i t a t i o n s inherent with p r o c e s s i n g the tens of m i l l i o n s of i n t e g r a l s at each i t e r a t i o n . I t i s easy to see why the CI step i s time consuming. Although the Fock matrix i s only n x n, the s i z e of the CI matrix goes more l i k e n ! Now the CI matrix i s very sparse, as indeed i t has to be, i f we are to get some of the eigenvectors and eigenvalues of matrices which can get to be as l a r g e as 10000x10000. Most of the e l e c t r o n i c s t r u c t u r e work at LASL i s concentrated on the CDC 7600 machines. We are c u r r e n t l y adapting our programs to use the CRAY machines e f f i c i e n t l y , but i t appears the CRAY's w i l l not be more than 10 times as powerful as a 7600. I t i s n ' t hard to t h i n k of problems which would overwhelm the CRAY's.



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Figure 1. Schematic representation of three areas of theoretical chemistry. The relationship between each of these areas and the modern supercomputer is discussed.

•

Let n = # atomic

Matrix

Size

Fock

n x n

2 e" f

CI Figure 2.

basis

functions

Limits 7600 CRAY-1 100

4

n /8

10»

4

10

10 4

~ n*xn

300

7

10

4

Computational requirements for molecular electronic structure calculations.


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In F i g . 3, we consider a few problems o f i n t e r e s t t o the s t r u c t u r e chemist i n the a r e a of t r a n s i t i o n metal chemistry. The molecules we consider here are of i n t e r e s t f o r t h e i r bonding p r o p e r t i e s and t h e i r e l e c t r o n i c a l l y e x c i t e d s t a t e s . The R e C l " i o n has a strong (almost quadruple) Re = Re bond — we estimate that we can do a f a i r job o f determining the energy of t h i s i o n with about 1/2 hour of CRAY time, using about 100 o r b i t a l s i n a s p l i t - v a l e n c e b a s i s . Not a l l the e l e c t r o n s i n t h i s system would be t r e a t e d e x p l i c i t y . At LASL, we r e g u l a r l y t r e a t l a r g e Z atoms u s i n g e f f e c t i v e core p o t e n t i a l s to e l i m i n a t e the innermost core e l e c t r o n s from the c a l c u l a t i o n . (1) Using about 250 o r b i t a l s , we can t r e a t t h i s mixed-valence ruthenium-pyrazine complex. This complex i s of i n t e r e s t because of i t s metal-organic bonding and the f a c t that the two ruthenium atoms are not equivalent to each other, even a t the Hartree-Fock l e v e l . Of i n t e r e s t a l s o i s t h i s bridged rhodium complex, which we estimate we can t a c k l e with about 350 o r b i t a l s . This molecule has the i n t e r e s t i n g property that, when d i s s o l v e d i n water, i t l i b e r a t e s hydrogen gas i n the presence of s u n l i g h t . This would be a very tough problem, even f o r the CRAY — we estimate 60 hours o f CRAY time to determine the s t r u c t u r e . One of the more r e l e v a n t d u t i e s of the s t r u c t u r e chemist i s to provide p o t e n t i a l energy surfaces to the chemical dynamicist. The p o t e n t i a l energy s u r f a c e i s determined by computing the e l e c t r o n i c energies of the molecular system as a f u n c t i o n o f the n u c l e a r geometry. In a d d i t i o n , i f s e v e r a l e l e c t r o n i c s t a t e s p a r t i c i p a t e i n the c o l l i s i o n dynamics, i t may a l s o be d e s i r a b l e to have a v a i l a b l e c e r t a i n matrix elements between e l e c t r o n i c s u r f a c e s . Now dynamicists tend to be rather demanding — at l e a s t by request i f not a l s o by need. Consequently, we a r r i v e at t h i s f i r s t law of p o t e n t i a l surface c a l c u l a t i o n s — the s t r u c t u r e chemist gets bored with running h i s program long before he can s a t i a t e the dynamicist. (Paraphrased from F i g . 4.) But look what happens — even i f the dynamicist compromises to the p o i n t that he s e t t l e s f o r 10 p o i n t s per nuclear degree of freedom, i t n e v e r t h e l e s s r e q u i r e s a bundle o f s t r u c t u r e c a l c u l a t i o n s to generate a s u r f a c e f o r a r e l a t i v e l y simple A+BC type r e a c t i o n . Now suppose we had a dynamicist who dared to study a four-body r e a c t i o n , l i k e AB+CD. Then imagine a s t r u c t u r e chemist w i l l i n g to compute a m i l l i o n p o i n t s on a p o t e n t i a l s u r f a c e . I t shouldn't be s u r p r i s i n g that there i s a t present only one p o t e n t i a l s u r f a c e which has been computed at enough p o i n t s and enough accuracy to s a t i s f y the dynamicist — the simplest of a l l n e u t r a l molecular systems — the H+H surface computed by Bowen L i u and Per Siegbahn. (2) Because of the high symmetry of t h i s system, they have c a l c u l a t e d a surface at about 250 p o i n t s (instead of the 1000 estimated).


2

?



WALKER E T AL.

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Chemistry

Estimated CRAY time (hrs)

Molecule 2

[[(NH )*Ru t - p y r a z i n e f 3

[Rh-(NC-C3H -CN) -RhP 6

0.5

104

RezCle "

4

+

252

20

340

60

'split-valence basis using effective potentials

Figure 3.

Examples of problems in transition-metal chemistry.

Law of Nature: Dynamicists will always want more points on a potential energy surface than one is willing to calculate.

Response of Electronic Structure Practitioners: Dynamic ists will usually settle for 10 points / degree of freedom.

Triatomics Tetratomics Figure 4.

Points 10 10 3

6

Potential energy surface calculations for the chemical dynamicist.


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Quantum Chemical Dynamics L e t ' s now turn our a t t e n t i o n to the requirements of the chemical dynamicist. Here we consider only quantum mechanical approaches t o chemical r e a c t i o n dynamics, and only mention that there a l s o e x i s t s a considerable computational technology which t r e a t s chemical dynamics by using c l a s s i c a l mechanics. The only type o f chemical r e a c t i o n we are l i k e l y to ever be able t o s o l v e r i g o r o u s l y i n a quantum mechanical way i s a threebody r e a c t i o n of the type A+BC -> AB+C. (See F i g . 5.) The input information to the dynamicist i s the p o t e n t i a l energy surface computed by the quantum s t r u c t u r e chemist. Given t h i s p o t e n t i a l s u r f a c e , we t r e a t the nuclear c o l l i s i o n dynamics using S c h r o d i n g e r s equation to model the chemical r e a c t i o n process. As was mentioned e a r l i e r , there i s only one f u l l y ab i n i t i o p o t e n t i a l energy s u r f a c e f o r chemical r e a c t i o n a v a i l a b l e to the dynamicist. This surface i s appropriate f o r an A+BC r e a c t i o n where A,B, and C are a l l three hydrogen atoms or hydrogen isotopes (H,D,T). F i g . 6 shows a contour map of the c o l l i n e a r part of t h i s s u r f a c e ( a l l three n u c l e i l i e on a s i n g l e l i n e ) ; the e s s e n t i a l f e a t u r e s o f the s u r f a c e topology a r e the entrance v a l l e y , the product v a l l e y , and the a c t i v a t i o n b a r r i e r separating these two v a l l e y s . Motion perpendicular to each v a l l e y corresponds to v i b r a t i o n o f the reactant or product molecule, and motion p a r a l l e l t o the f l o o r of the v a l l e y measures progress of the r e a c t i o n , from r e a c t a n t s to products. The c l a s s i c a l mechanical s o l u t i o n t o chemical r e a c t i o n dynamics i s accomplished i n f a c t by s o l v i n g f o r the motion of a p o i n t mass p a r t i c l e on t h i s hypersurface. Reaction corresponds to a t r a j e c t o r y which s t a r t s out i n the reactant v a l l e y , crosses the b a r r i e r , and ends moving out i n t o the product v a l l e y . Quantum mechanically, the r e a c t i v e dynamics i s expressed i n a more wavelike language. By s o l v i n g Schrodinger's equation, we t r e a t the problem where an i n i t i a l p r o b a b i l i t y wave of reactants i s sent i n towards the a c t i v a t i o n b a r r i e r from r e a c t a n t s . When the wave h i t s the b a r r i e r , part o f i t i s r e f l e c t e d and part o f i t i s transmitted. The r e f l e c t e d part of the wave corresponds to non-reactive c o l l i s i o n events, and the transmitted part corresponds t o r e a c t i o n . The a c t u a l equations we solve are c a l l e d the close-coupled equations. (See F i g . 7.) They are obtained from the Schrodinger equation i n the f o l l o w i n g way: (1) we f i r s t d e f i n e a l l but one o f the coordinates o f the system to be " t a r g e t " coordinates and the f i n a l coordinate i s c a l l e d the " s c a t t e r i n g c o o r d i n a t e " or " r e a c t i o n c o o r d i n a t e . " The r e a c t i o n coordinate t e l l s us where we are i n our journey along the p o t e n t i a l surface from the reactant v a l l e y towards the product v a l l e y . Basis f u n c t i o n s are defined which d e s c r i b e motion i n a l l the t a r g e t coordinates. These b a s i s f u n c t i o n s a r e square i n t e g r a b l e f o r the target coordinate degrees of freedom, but the f u n c t i o n which describes motion i n the r e a c t i o n coordinate i s determined n u m e r i c a l l y . The equations f


3.

WALKER ET AL.


•

Theoretical

Chemistry

A + B-C -

A B C -> A - B + C

•

Requires potential energy surfacets) from electronic structure calculations

•

Solve Schrodinger equation for dynamics

Figure 5.

Quantum chemical dynamics. Scope and method of currently tractable problems.

Figure 6.

Contour map of H-\-H

t

• •

•

Figure 7.

collinear chemical potential energy surface.

Separate all (3N-3) coordinates into one scattering coordinate and (3N-3)-l internal coordinates Expand wave function using square integrable basis functions for (3N-3)-l coordinates and solve numerically for function of scattering coordinate. Leads to a set of coupled linear second order differential equations. One equation for each "channel." How close coupled equations are obtained in chemical dynamics problems.


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f o r these s c a t t e r i n g f u n c t i o n s are the close-coupled equations. These equations are a set of coupled second order l i n e a r o r d i n a r y d i f f e r e n t i a l equations. The d i f f i c u l t y i n s o l v i n g problems i n quantum chemical dynamics i s simply t h i s — how many coupled equations are there? The answer i s that there i s one equation f o r every "channel" i n the c l o s e c o u p l i n g expansion. Each channel i s defined by a unique set of quantum numbers f o r the target degrees o f freedom. There are f i v e such l a b e l s f o r each channel. They are (1) J — the t o t a l angular momentum and (2) M, i t s p r o j e c t i o n on an axis f i x e d i n space. In a d d i t i o n there are l a b e l s (3) n f o r the v i b r a t i o n a l motion of the molecule, (4) j f o r the molecular r o t a t i o n a l degree of freedom, and (5) I f o r the atom-molecule o r b i t a l angular momentum. The equations f o r one s e t of (J,M) are uncoupled from equations f o r other values of (J,M). The equations f o r a f u n c t i o n l a b e l e d by one value of (n,j,&) are coupled to values of a l l the other f u n c t i o n s l a b e l e d by (the same or) d i f f e r e n t values of (n,j,&). The number of coupled equations we have to solve t h e r e f o r e depends on the number of molecular v i b r a t i o n - r o t a t i o n s t a t e s we have to t r e a t i n the s c a t t e r i n g dynamics at each c o l l i s i o n energy. In the next paragraph, we present a rudimentary look a t the algorithm we use to s o l v e these coupled equations. This method i s c a l l e d R-matrix propagation; (3) and although there are s e v e r a l other methods e q u a l l y capable of s o l v i n g the coupled equations, we use R-matrix propagation as an example because i t i l l u s t r a t e s the k i n d o f computer algorithms we r e q u i r e . The R-matrix i t s e l f contains the s c a t t e r i n g information we need; the f i n a l R-matrix i s assembled i n a r e c u r s i v e f a s h i o n using the a n a l y t i c s o l u t i o n of the s c a t t e r i n g problem over a small region o f the s c a t t e r i n g coordinate. The algorithm works i n the f o l l o w i n g way: given (1) an o l d R-matrix a s s o c i a t e d with the s o l u t i o n of the s c a t t e r i n g problem over one r e g i o n of space; and given (2) a s e c t o r R-matrix which d e f i n e s the s c a t t e r i n g s o l u t i o n over a small incremental r e g i o n o f space, we can (3) assemble a new R-matrix which i s a s s o c i a t e d w i t h the s o l u t i o n of the s c a t t e r i n g problem over the ( o l d + incremental = new) r e g i o n of space. The r e c u r s i o n equation i s a matrix equation o f order n,

h

'

In

+

(

+

H2i 5i Hii

r l

Ei2

where there a r e n channels i n the c l o s e - c o u p l i n g expansion of the wavefunction. As you can see, t h i s r e c u r s i o n formula i n v o l v e s very standard matrix operations — m u l t i p l i c a t i o n and i n v e r s i o n . The a n a l y t i c s o l u t i o n of the coupled equations i n the incremental r e g i o n i s d e f i n e d i n terms of the eigenvalues and eigenvectors of the c o u p l i n g matrix. So you can see that the b a s i c numerical algorithm we r e q u i r e our supercomputer to handle e f f e c t i v e l y are standard matrix operations — m u l t i p l i c a t i o n , d l a g o n a l i z a t i y i , and i n v e r s i o n . A l l these algorithms go a s y m p t o t i c a l l y as n — and so the complexity of the quantum d y n a m i c i s t s problem i s measured (as we s a i d p r e v i o u s l y ) by the s i z e (n) of the c l o s e coupled equations. T



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So l e t ' s r e t u r n our a t t e n t i o n again to the question o f the s i z e o f the coupled equations and consider some examples. Many of the chemical r e a c t i o n s we are i n t e r e s t e d i n are dominated by an a c t i v a t i o n b a r r i e r which separates the reactant and product v a l l e y s o f the p o t e n t i a l energy hypersurface. (See F i g . 8.) The energy o f t h i s a c t i v a t i o n b a r r i e r l o c a t e s the general energy range of i n t e r e s t t o the r e a c t i o n dynamicist — because there i s n ' t very much r e a c t i o n a t energies below the b a r r i e r h e i g h t , where only quantum t u n n e l l i n g processes can c o n t r i b u t e to r e a c t i o n . But, as we show s c h e m a t i c a l l y i n F i g . 8, there may be s e v e r a l molecular energy s t a t e s below the a c t i v a t i o n b a r r i e r . A l l these s t a t e s , a t the very l e a s t , must be i n c l u d e d i n the c l o s e c o u p l i n g expansion. L e t ' s now consider s e v e r a l examples. The simplest o f a l l r e a c t i o n s i s the H+H r e a c t i o n . The H v i b r a t i o n a l l e v e l s are f a i r l y widely spaced, but we must a l s o i n c l u d e the r o t a t i o n a l manifold o f l e v e l s a s s o c i a t e d with each v i b r a t i o n a l l e v e l . (See F i g . 9.) Now, i t i s t h i s r o t a t i o n a l manifold of l e v e l s (and the degeneracies o f s t a t e s a s s o c i a t e d with each v i b r a t i o n - r o t a t i o n l e v e l ) which u l t i m a t e l y breaks the bank i n the s i z e o f the c l o s e c o u p l i n g expansion. In order to t r e a t quantum dynamical problems, i t w i l l be necessary to introduce approximations which reduce the s i z e of the set o f coupled equations. Two promising approximations are the c e n t r i f u g a l sudden (CS) (4^ 5) approximation and the i n f i n i t e order sudden (IOS) approximation.(5, 6) The CS approximation removes the c o u p l i n g between the j and I angular momenta, thereby reducing the s i z e o f the coupled equations from n to approximately n . In t h i s approximation, each (n,j) energy l e v e l generates only one channel instead o f (2j+l) channels. The more d r a s t i c IOS approximation appears to be promising f o r systems i n which the molecular species r o t a t e s very slowly on the s c a l e of the c o l l i s i o n time. This approximation removes i n e f f e c t a l l the r o t a t i o n a l l e v e l s from the system. For the H+H^ system, we estimate that we can j u s t about s o l v e t h i s e a s i e s t o f a l l problems with current s t a t e - o f - t h e - a r t computers a t the 100-channel l e v e l . I f we can use the CS approximation f o r t h i s system, we can i n f a c t go to q u i t e high s c a t t e r i n g energies. But remember that H+H i s the simplest of a l l r e a c t i o n s . Moving more i n the d i r e c t i o n of true chemistry, consider next a r e a c t i o n f o r which only two n u c l e i are hydrogens (instead o f three) — the F+H r e a c t i o n . This r e a c t i o n i s over 1 eV exothermic i n going from the reactant v a l l e y , over a small (1 k c a l ) b a r r i e r , t o the product v a l l e y . The exothermicity o f r e a c t i o n means that there are s e v e r a l e n e r g e t i c a l l y a c c e s s i b l e (open) v i b r a t i o n a l channels f o r t h i s system even at the t h r e s h o l d for reaction. I f we i n c l u d e a l l the r o t a t i o n a l l e v e l s with each v i b r a t i o n , and the proper (2j+l) r o t a t i o n a l degeneracies, we have an unthinkably l a r g e number of coupled equations to solve — over 1200 channels. (See F i g . 10.) To s o l v e t h i s problem, we must 2

2

2

2




How

•

M a n y

Coupled

Channels

a r e

There?

Reactions are dominated by activation barriers

=

Activation Barrier

Reactants Products REACTION COORDINATE •

Need all open channels, some closed

•

State of art =

channels

100 channels (CRAY = 300)

Figure 8. Schematic representation of chemical potential energy surfaces. Counting of states below reaction barrier for both reactants and products gives a minimal estimate of numbers of coupled equations to be solved.


WALKER E T AL.

Theoretical


H + H

2

Chemistry

Energy Levels

Figure 9. Counting of channels for the H reaction. Reaction barrier is at 0.4 eV; state-of-the-art calculations are performed to slightly above 1 eV. Arrows are drawn whenever another 100 coupled channels are required. s

F + H

2

Energy Levels

Figure 10. Counting channels for the FH reaction. Conditions are as in Figure 9, except that arrows count states for CS approximation (each vibration-rotation level counts only once). Reaction threshold is at 1.65 eV. t



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use approximations such as the CS approximation, which reduces the problem to the much more manageable 100-channel l e v e l . For our f i n a l quantum dynamics example, consider what happens when we s u b s t i t u t e a l i t h i u m atom f o r one of the remaining hydrogens — the Li+FH r e a c t i o n . (See F i g . 11.) This semie m p i r i c a l p o t e n t i a l s u r f a c e ( c o l l i n e a r ) shows a narrow entrance channel v i b r a t i o n a l v a l l e y , a shallow w e l l i n the entrance channel, a b a r r i e r , and a broad product v i b r a t i o n a l v a l l e y . Even u s i n g the CS approximation, the energy l e v e l diagram f o r t h i s r e a c t i o n makes t h i s problem a c c e s s i b l e only to the f u l l power of a CRAY l e v e l machine. Anyone f o o l i s h enough to t a c k l e the problem r i g o r o u s l y w i l l have to face a 10000-channel system at energies j u s t above t h r e s h o l d ! The moral of our s t o r y of the quantum chemical r e a c t i o n dynamicist should be p e r f e c t l y c l e a r — at l e a s t two hydrogens are the dynamicists best f r i e n d . Indeed, our current supercomputers may seem to be a b i t l e s s super. Quantum O p t i c s The i n t e r a c t i o n of molecules with electromagnetic r a d i a t i o n i s of fundamental i n t e r e s t to the chemist. When the e l e c t r o magnetic f i e l d i s r e l a t i v e l y weak, we can d e s c r i b e these i n t e r a c t i o n s u s i n g p e r t u r b a t i o n theory. The study of s i n g l e photon t r a n s i t i o n s induced between molecular states by weak f i e l d s i s the province of the molecular s p e c t r o s c o p i s t . But now, with the ever more powerful r a d i a t i o n f i e l d s a v a i l a b l e from l a s e r technology, we are i n a p o s i t i o n to study the i n t e r a c t i o n between molecules and electromagnetic r a d i a t i o n at i n t e n s i t i e s too l a r g e f o r p e r t u r b a t i o n methods to work. The somewhat broader f i e l d of quantum o p t i c s seeks to d e s c r i b e the time e v o l u t i o n of molecules i n the presence of these intense f i e l d s . Of course, before we can f o l l o w the m i g r a t i o n of energy among the v a r i o u s degrees of freedom of a ( p o s s i b l y l a r g e ) molecule, we must f i r s t know what the e l e c t r o n i c , v i b r a t i o n a l , and r o t a t i o n energy states of the molecule are i n the absence of any r a d i a t i o n f i e l d . The e f f e c t of the f i e l d i s to move p o p u l a t i o n from the i n i t i a l molecular s t a t e i n t o other molecular s t a t e s i n a time dependent way. The s o l u t i o n of t h i s problem can be obtained by s o l v i n g the timedependent Schrodinger equation, so long as the molecule we are studying i s modeled at zero pressure. At f i n i t e pressures (when c o l l i s i o n s are present) the Schrodinger p i c t u r e i s too d i f f i c u l t to s o l v e d i r e c t l y ; i n t h i s case, we can model the incoherent (phase destroying) e f f e c t s of c o l l i s i o n s upon the coherent e x c i t a t i o n induced by the electromagnetic f i e l d by r e s o r t i n g to a Bloch equation (or d e n s i t y matrix) formalism. C o l l i s i o n s are modelled by decay r a t e s not only i n the diagonal (but a l s o the o f f - d i a g o n a l ) terms of the d e n s i t y matrix. We a l s o have one f u r t h e r c o n s t r a i n t i n developing methods to t r e a t problems i n quantum o p t i c s — because l a s e r pulses l a s t f o r a r e l a t i v e l y long time i n comparison to the time a s s o c i a t e d with molecular v i b r a t i o n



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Figure 11. Counting channels for the LiFH reaction. As in Figures 9 and 10, arrows count states for CS approximation. Reaction threshold is near 0.6 eV.


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and r o t a t i o n , we must s o l v e our time-dependent Schrodinger or Bloch equation i n a way which gives answers e f f i c i e n t l y f o r long times. Consider f o r a moment the Bloch equation f o r the d e n s i t y matrix, p,


i p = pH - Hp - iTp The Bloch equation gives the time d e r i v a t i v e of the d e n s i t y matrix p i n terms of i t s commutator with the Hamiltonian f o r the system, and the decay r a t e matrix T. Each of the m a t r i c e s , p, H, and T are n x n matrices i f we consider a molecule with n v i b r a t i o n r o t a t i o n s t a t e s . We so^ve t h i s equation by r e w r i t i n g the n x n square matrix p as an n -element column v e c t o r . R^writ^ng p i n t h i s way transforms the H and T matrices i n t o an n x n complex general matrix R. We o b t a i n P = Rp The s o l u t i o n of the transformed equation i s obtained by exponent i a t i n g t h i s R matrix. To e f f i c i e n t l y exponentiate t h i s matrix we must f i r s t ~ d i a g o n a l i z e i t , exponentiate the eigenvalues, and back transform with the eigenvectors. This back transformation procedure i s repeated f o r every time at which we wish to know the molecular p o p u l a t i o n s . A t y p i c a l problem of i n t e r e s t at Los Alamos i s the s o l u t i o n of the i n f r a r e d m u l t i p l e photon e x c i t a t i o n dynamics of s u l f u r h e x a f l u o r i d e . T h i s very problem has been q u i t e popular i n the l i t e r a t u r e i n the past few years. (7) The s o l u t i o n of t h i s problem i s modeled by a molecular Hamiltonian which e x p l i c i t l y t r e a t s the asymmetric s t r e t c h v.. ladder of the molecule coupled i m p l i c i t l y to the other molecular degrees of freedom. (See F i g . 12.) We consider the the f i r s t seven v i b r a t i o n a l s t a t e s of the mode of SF, (6v~); the o c t a h e d r a l symmetry of the SF^ molecule makes these v i b r a t i o n a l l e v e l s degenerate, and c o u p l i n g between v i b r a t i o n a l and r o t a t i o n a l motion s p l i t s these degeneracies slightly. Furthermore, there i s a r o t a t i o n a l manifold of s t a t e s a s s o c i a t e d with each v i b r a t i o n a l l e v e l . Even to d e s c r i b e the zeroth-order l e v e l s t a t e s of t h i s molecule i s i t s e l f a f a i r l y complicated problem. Now i f we were to i n c l u d e c o l l i s i o n s i n our model of m u l t i p l e photon e x c i t a t i o n of SF^, we wou^d have to s o l v e a m a t r i x Bloch equation with a minimum of 84 x 84 elements. C l e a r l y such a problem i s beyond our current a b i l i t i e s , so i n f a c t we n e g l e c t c o l l i s i o n a l e f f e c t s i n order to stay with a Schrodinger p i c t u r e of the e x c i t a t i o n dynamics. In the Schrodinger p i c t u r e , we can i n c l u d e the diagonal elements of the Y matrix, which model the c o u p l i n g of the e x p l i c i t l y t r e a t e d V^-laSder s t a t e s with the other i m p l i c i t l y t r e a t e d molecular s t a t e s . The exponentiation of the c o u p l i n g matrix i n the Schrodinger p i c t u r e r e q u i r e s the d i a g o n a l i z a t i o n of an n x n complex general matrix. Populations at s e v e r a l times are



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S F + Nhi/ + M — 6

SF

6

f

+M

# Asymmetric stretch ( i / ) ladder dynamics 3

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Vibrational degeneracy is .5(N+lXN+2)

•

Include up to 6i/ in H, get 84 x84^ matrix to diagonalize 3

2

Figure 12. Schematic of multiple photon excitation dynamics of SF . Groups of levels show lowest three v vibrational states. Higher states are split by rotational interactions with vibrational motion. 6

s



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computed by the back transformation method, and a quadrature over time of those populations gives the leakage of amplitude i n t o the SF^ quasicontinuum degrees of freedom. This whole process i s repeated f o r each new i n i t i a l r o t a t i o n a l s t a t e J , l a s e r frequency v, and l a s e r i n t e n s i t y I. Our c a l c u l a t i o n s at LASL can r e q u i r e up to 10 hours of 7600 time f o r each l a s e r power of interest. There are s e v e r a l other i n t e r e s t i n g t o p i c s i n quantum o p t i c s which we would l i k e to be able to study. For example, we would l i k e model problems i n double resonance spectroscopy, where there are two electromagnetic f i e l d s with p o s s i b l y d i f f e r e n t p o l a r i z a t i o n s simultaneously i n t e r a c t i n g with a molecule. This problem resembles the m u l t i p l e photon e x c i t a t i o n problem i n that there i s p o p u l a t i o n m i g r a t i o n along ladders of s t a t e s , but i n t h i s case there can be a v a s t l y l a r g e r number of quantum l e v e l s to t r e a t — on the order of 2(2J+1). At room temperature, the most probable value of J f o r S F i s about 60, which implies a 250 s t a t e calculation. F i n a l l y , we a l s o mention a s u b s t a n t i a l l y more complex problem — that of l a s e r pulse propagation through an absorbing medium. In t h i s case we are asking not only what happens to the molecule i n the presence of an electromagnetic f i e l d , but a l s o what happens m a c r o s c o p i c a l l y to the f i e l d i n the presence of the molecule. The s o l u t i o n of t h i s problem r e q u i r e s t r e a t i n g the m u l t i p l e photon dynamics problem s e l f - c o n s i s t e n t l y with a s o l u t i o n of Maxwell's equation over a g r i d of p o i n t s i n space. 6

Conclusions In summary, our i n t e n t i o n has been to give examples of the types of problems we are i n t e r e s t e d i n at LASL. Our a p p e t i t e f o r computationally d i f f i c u l t problems has not been d u l l e d by the current a v a i l a b i l i t y of computer resources. In the area of molecular e l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s , we need computers f o r which there can be w r i t t e n e f f i c i e n t algorithms to d i a g o n a l i z e l a r g e m a t r i c e s , and i n the case of CI c a l c u l a t i o n s , we need e f f i c i e n t i n d i r e c t addressing c a p a b i l i t i e s ( g a t h e r - s c a t t e r operations) i n order to process these matrices whose elements are 99% zeroes. E i t h e r we need e f f i c i e n t 10 c a p a b i l i t i e s i n order to process the l i s t s of m i l l i o n s of i n t e g r a l s , or i t has to be cheaper to c a l c u l a t e these i n t e g r a l s as we go along. In the area of quantum dynamics, we need again computers capable of e f f i c i e n t l y performing standard types of matrix operations ( i n v e r s i o n , d i a g o n a l i z a t i o n , m u l t i p l i c a t i o n ) on l a r g e matrices of the order of s e v e r a l hundreds. And i n the area of quantum o p t i c s , we need s i m i l a r types of c a p a b i l i t i e s - standard matrix manipulations - but now our matrices are complex general i n s t e a d of r e a l symmetric. In each of the f i e l d s discussed here, s t a t e - o f - t h e - a r t c a l c u l a t i o n s r e q u i r e the f u l l c a p a b i l i t i e s of modern computers.


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Newer supercomputers will need to be several orders of magnitude more powerful to efficiently attack many of the problems currently facing the theoretical chemist.


Literature Cited 1.

Hay, P. J.; Wadt, W. R.; Kahn, L. R.; Bobrowicz, F. W. J. Chem. Phys. 1978, 69, 984.

2.

Liu, B. J. Chem. Phys. 1973, 58, 1925. Chem. Phys. 1978, 68, 2457.

3.

Light, J. C.; Walker, R. B. J. Chem. Phys. 1976, 65, 4272. Stechel, E. B.; Walker, R. B.; Light, J. C. J. Chem. Phys. 1978, 69, 3518. Light, J. C.; Walker, R. B.; Stechel, E. B.; Schmalz, T. G. Comput. Phys. Comm. 1979, 17, 89.

4.

Pack, R. T J. Chem. Phys. 1974, 60,633.McGuire P., Kouri, D. J. J. Chem. Phys. 1974, 60, 2488.

5.

Kouri, D. J. in "Atom-Molecule Collision Theory: A Guide for the Experimentalist," ed. R. B. Bernstein; Plenum Press: New York, 1979; p. 301-358.

6.

Khare, V.; Kouri, D. J.; Baer, M. J. Chem. Phys. 1979, 71, 1188. Barg, G.; Drolshagen, G. Chem. Phys. 1980, 47, 209. Bowman, J. M.; Lee, K. T. J. Chem. Phys. 1980, 72, 5071.

7.

Galbraith, H. W.; Ackerhalt, J. R. in "Laser Induced Chemical Processes," ed J. I Steinfeld; Plenum Press:New York, to appear.

Siegbahn, P.; Liu, B.

RECEIVED August 4, 1981.


Supercomputers in Chemistry - American Chemical Society

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