The Use of Vector Processors in Quantum Chemistry - ACS Publications

1 The Use of Vector Processors in Quantum Chemistry Experience i n the U n i t e d K i n g d o m M A R T Y N F . G U E S T and S T E P H E N

WILSON

Downloaded by 80.82.77.83 on April 3, 2018 | https://pubs.acs.org Publication Date: November 6, 1981 | doi: 10.1021/bk-1981-0173.ch001

Science and Engineering Research Council, Daresbury Laboratory, Daresbury, Warrington W A 4 4 A D , U . K .

The purpose of this paper is to review the impact which vector-processing computers(1,2) are having on ab initio quantum chemical calculations giving special emphasis to experience gained in the United Kingdom. The advent of such powerful computational tools is having, and will continue to have, an important influence on computational quantum chemistry. Calculations which were very time consuming are becoming routine; calculations which were impossible are now tractable. This review is necessarily selective, and is divided into several sections. Initially we give an overview of the availability of supercomputers in the U.K., and summarise the experience gained in the implementation of various quantum chemistry packages. Optimization of Quantum Chemistry codes on the CRAY-1 is considered, with integral evaluation, self-consistent-field and integral transformation phases of quantum chemical studies being considered together with some aspects of the correlation problem. The significant impact of the CRAY-1 in several areas of electronic structure research is then outlined, with particular attention given to the evaluation of the components of electron correlation energy which may be associated with higher order excitations and to the development of basis sets suitable for accurate studies. This is followed by some concluding remarks. Supercomputers in the United Kingdom The CPAY-1 vector processing computer at the Science Research Council's (S.E.R.C) Daresbury Laboratory, is at the centre of a network providing large scale computational facilities for Universities in the United Kingdom. This is the only supercomputer available at present to Quantum Chemists in the U.K., and this article will therefore be restricted to experience gained on the CRAY-1, although this experience will undoubtedly be relevant to future applications on machines such as the ICL Distributed Array Processor (DAP) (see reference (2) for a detailed description) and the CDC Cyber 203/205. 0097-6156/81 /0173-0001 $09.50/0 © 1981 American Chemical Society

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

2

SUPERCOMPUTERS IN

CHEMISTRY


The main c h a r a c t e r i s t i c s of the CRAY-1 computer are shown i n Table I (see a l s o reference (2^)). The s c a l a r operations are seen to be approximately twice that of the CDC 7600 and IBM 360/195. The maximum vector c a p a b i l i t y occurs f o r matrix m u l t i p l i c a t i o n , for which the measured time on the CRAY-1 i s twenty times f a s t e r than the best hand coded r o u t i n e s on the CDC 7600 or IBM 360/195. The maximum rate i s c i r c a . 135 Mflops ( M i l l i o n s of f l o a t i n g - p o i n t operations per second) f o r matrices t h a t have dimensions which are a m u l t i p l e of 64, the vector r e g i s t e r s i z e . The r a t e of computat i o n f o r matrix m u l t i p l i c a t i o n i s shown i n f i g u r e 1 as a f u n c t i o n of matrix s i z e . The Daresbury Laboratory CRAY-1 computer i s accessed by means of an IBM 370/165 which i s l i n k e d to computers at the S.E.R.C's Rutherford Laboratory, C.E.R.N, and workstations i n many U n i v e r s i ties. The S.E.R.C. network i n f a c t incorporates l i n k s to 10 mainframe and 76 minicomputers and to 44 d i f f e r e n t s i t e s . The CRAY-1 was i n s t a l l e d at Daresbury f o r an i n i t i a l p e r i o d of two years, extendable f o r a t h i r d year. The S.E.R.C. buys an average of e i g h t hours per day from CRAY Research Inc. L t d . , and the p o s s i b i l i t y e x i s t s t h a t the machine w i l l be upgraded to a CRAY-1 Model S/500. Proposals are also under c o n s i d e r a t i o n f o r the i n s t a l l a t i o n of supercomputers a t the two l a r g e s t u n i v e r s i t y r e g i o n a l computer cent r e s - London and Manchester - and at the S.E.R.C's Rutherford Laboratory where the e x i s t i n g twin IBM 360/195 machines are scheduled f o r replacement i n 1982/3. Again these machines would be a c c e s s i b l e v i a workstations i n a number of U n i v e r s i t y departments around the U.K. The Daresbury Laboratory has a p p l i e d the CRAY-1 to i t s m u l t i faceted s c i e n t i f i c environment since June 1979. D i s c i p l i n e s benef i t t i n g from the a v a i l a b i l i t y are numerous, with a b r i e f summary of the s c i e n t i f i c research i n v o l v e d i n the p e r i o d June 1979 - June 1980 being given i n Table I I . In t h i s t a b l e we a l s o present some reported improvements determined on v e c t o r i s a t i o n of various packages. Much use of the CRAY-1 computer has been made by the S.E.R.C's C o l l a b o r a t i v e Computational P r o j e c t s (C.C.P.s). These p r o j e c t s aim to co-ordinate the development of s o p h i s t i c a t e d s o f t ware i n various f i e l d s of research w i t h i n the U.K. The f i r s t of these p r o j e c t s i s concerned with e l e c t r o n c o r r e l a t i o n e f f e c t s i n molecules, and i s of p a r t i c u l a r i n t e r e s t to Quantum Chemists. Implementation and Performance of Quantum Chemistry Packages. A Preliminary Investigation Perhaps the f i r s t question to be considered i n contemplating the use of a vector processor i n Quantum Chemistry (QC) i s j u s t how much advantage i s o b t a i n a b l e with the minimum amount of e f f o r t i . e . by simply implementing software from a s c a l a r machine with l i t t l e or no m o d i f i c a t i o n . The answer to t h i s question i s r e a d i l y obtainable by benchmarking the machine against some standard on a v a r i e t y o f widely used QC packages. Such an e x e r c i s e would shed


1.

GUEST AND WILSON

Table I:

Vector Processors in Quantum

Chemistry

The Main C h a r a c t e r i s t i c s of the CRAY-1• Facts and Figures (from "The CRAY-1 Computer System", P u b l i c a t i o n No.2240008B, 1977, Cray Research Inc.)

CPU


Instruction size Repertoire s i z e Clock p e r i o d Instruction stack/buffers Functional units

Programmable r e g i s t e r s

Max.

vector r e s u l t r a t e

16 or 32 b i t s 128 i n s t r u c t i o n codes 12.5 nsec 64 words (4096 b i t s ) twelve: 3 integer add 1 integer m u l t i p l y 2 shift 2 logical 1 f l o a t i n g add 1 floating multiply 1 r e c i p r o c a l approx. 1 population count 8 x 64 64-bit 73 64-bit 72 24-bit 1 7-bit 12.5 nsec/unit

FLOATING POINT COMPUTATION RATES ( r e s u l t s per second) Addition 80 x 10^/sec Multiplication 80 x 10 /sec Division 25 x 10 /sec 6

6

MEMORY Technology Word length Address space Data path width ( b i t s ) Cycle time Size

Organisation/interleave Maximum band width

b i p o l a r semiconductor 72 b i t s (64 data, 8 SECDED) 4M words 64 (1 word) 50 nsec• 262,144 words or 524,288 words or 1,048,576 words 16 banks (8 banks o p t i o n a l ) 80 x 10 words/sec (5.1 x 10 b i t s / s e c ) SECDED 6

9

Error

checking


3

4

SUPERCOMPUTERS IN CHEMISTRY


15CH

Matrix dimension Figure 1.

Table I I :

Matrix multiplication timing. Execution rate, in MFLOPS, as a function of vector length.

is plotted

Summary of the R e l a t i v e Performance of the CRAY-1 i n S c i e n t i f i c Research a t the S.R.C. Daresbury Laboratory. R e l a t i v e Performance*

Research Area

Nuclear Physics Astronomy Protein Crystallography Oceanography Atomic & Molecular Physics Quantum Chemistry Plasma Physics P h y s i c a l Chemistry Surface Physics

Computer used as Benchmark

After Modification No Modifications

Total

Selected Routines 80 26

IBM 370/165 CDC 7600 ICL 2980

10 2.5 14

50 9.3

IBM 370/165 IBM 370/165

11 4-6

8-34 16-30

IBM CDC CDC IBM DEC CDC

370/165 7600 7600 370/165 10 7600

2-10 (E)

E,

E*(pf>)

E(hp)

E(hh)

3

s

basis

FH CfcH OH N CO SiS 2

2

23 23 29 34 34 34

(b)

t

D

3.14 2.92 15.51 14.93 18.32 18.55

t r

(b) T

41.52 41.53 107.71 352.49 339.00 352.58

(b)

t

Q 1.80 1.76 5.59 9.92 10.30 10.30

(a) a c t i v e o r b i t a l s (b) seconds



28


C a l c u l a t i o n s have, unfortunately shown that the t r i p l e - e x c i t a t i o n component of the c o r r e l a t i o n energy i s chemically s i g n i f i cant, t h a t i s greater than one m i l l i h a r t r e e . For example, f o r the water molecule, using the b a s i s set of t h i r t y - n i n e S l a t e r funct i o n s given by Rosenberg and S h a v i t t (42), a fourth-order t r i p l e e x c i t a t i o n energy of -7.9 m i l l i h a r t r e e was obtained (39). The t r i p l e - e x c i t a t i o n energy f o r a number of small atoms and molecules i s shown i n Table VII (48). I t can be seen from the Table t h a t t h i s component of the c o r r e l a t i o n energy i s q u i t e l a r g e and p a r t i c u l a r l y so f o r multiply-bonded systems, such as the nitrogen molecule. C a l c u l a t i o n s on p o r t i o n s of the p o t e n t i a l energy curves of some m u l t i p l y bonded diatoms have shown that the t r i p l e - e x c i t a t i o n component of the c o r r e l a t i o n energy v a r i e s q u i t e markedly with nuc l e a r geometry. Furthermore, the r e s u l t s shown i n Table VIII f o r the water molecule using d i f f e r e n t b a s i s sets show t h a t t h i s energy has a somewhat stronger dependence on the q u a l i t y of the b a s i s set than other components of the c o r r e l a t i o n energy (4J_) • ( i i ) Basis sets f o r accurate molecular c a l c u l a t i o n s . Fundamental to almost a l l a p p l i c a t i o n s of quantum mechanics to molecules i s the use of a f i n i t e b a s i s s e t . Such an approach leads to computational problems which are w e l l s u i t e d to v e c t o r i s ation. For example, by using a b a s i s set the i n t e g r o - d i f f e r e n t i a l Hartree-Fock equations become a set of a l g e b r a i c equations f o r the expansion c o e f f i c i e n t s - a set of matrix equations. The absolute accuracy 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 i s u l t i mately determined by the q u a l i t y of the basis set employed. No amount of c o n f i g u r a t i o n i n t e r a c t i o n w i l l compensate f o r a poor choice of basis s e t . The supercomputers open up the p o s s i b i l i t y of using much l a r ger b a s i s sets i n routine c a l c u l a t i o n s and thus achieving greater p r e c i s i o n than i s c u r r e n t l y a t t a i n a b l e . A r e c e n t l y developed concept which w i l l enable large b a s i s s e t s to be used i n molecular c a l c u l a t i o n s i s the U n i v e r s a l Basis Set (43-47). To recover a s i g n i f i c a n t f r a c t i o n of the c o r r e l a t i o n energy i t i s u l t i m a t e l y necessary to use a moderately l a r g e b a s i s set. Such a large basis set has a considerable degree of f l e x i b i l i t y and can t h e r e f o r e be t r a n s f e r r e d from system to system without regard to the p a r t i c u l a r atoms involved and with l i t t l e l o s s i n accuracy. I h i s approach has been demonstrated using a univers a l even-tempered basis set i n which the o r b i t a l exponents are d e f i n e d by a geometric s e r i e s

where I denotes the symmetry type of the basis f u n c t i o n s . Values of a and 3 have been determined which are capable of g i v i n g an accurate d e s c r i p t i o n of f i r s t - r o w and second-row atoms within the Hartree-Fock model. Using the f o l l o w i n g parameters:


1.

GUEST AND WILSON

Table VTI


Chemistry

29

Fourth-order linked-diagram t r i p l e - e x c i t a t i o n and quadruple e x c i t a t i o n components of the e l e c t r o n c o r r e l a t i o n energy i n a number of atoms and small molecules ^ a


System

Ne Ar BH FH A*H ClH

BeH2 0H2 MgH2 SH2 BH3

CH

+ 3

NH (D h) NH (C ) OH (D h) OH (C ) A*H PH (D ) PH (C ) 3

3

3

3v

+

3

3

+

3

3 v

3

3

3 h

3

3 v

N2

CO BF CS SiO SiS

E

SCF

E

-128.54045 -526.80194 -25.12890 -100.06007 -242.45553 -460.09456 -15.77024 -76.05558 -200.72028 -398.70077 -26.39834 -39.24533 -56.20957 -56.21635 -76.33276 -76.33475 -243.63770 -342.41887 -342.47717 -108.97684 -112.77551 -124.15642 -435.33679 -363.82790 -686.48488

E

[2/1]

E 4T

-2.427 -1.159 -0.770 -4.341 -0.521 -2.479 -0.285 -5.539 -0.084 -3.383 -1.327 -1.588 -4.677 -5.040 -4.573 -4.778 -0.705 -3.145 -3.064 -17.602 -15.596 -8.392 -18.534 -16.700 -11.719

-210.034 -161.846 -86.731 -222.427 -72.745 -170.724 -68.623 -223.457 -63.105 -173.182 -121.471 -134.462 -208.747 -210.245 -221.592 -223.315 -105.091 -168.516 -167.680 -317.938 -295.510 -254.719 -260.619 -275.312 -225.190

0.415 2.068 1.290 1.071 1.116 2.789 0.699 2.018 0.586 3.232 1.562 1.691 2.450 2.591 2.013 2. 128 1.240 3.173 2.998 6.284 4.536 2.088 7.360 4.078 6.436

)The s e l f - c o n s i s t e n t - f i e l d energies are given i n hartree; the t r i p l e - e x c i t a t i o n energy, E i * , the q u a d r u p l e - e x c i t a t i o n energy, Eif , and the [2/1] Pad* approximant, E . , are given i n millihartree. ' T

Q

1

J



Electronic

(b) c o n t r a c t i o n

-4.860 -5.320 -6.148 -7.934

+1.425 +0.157 +2.158 +3.059

-10.691 -7.985 -12.451 -14.578

-4.963 -5.043 -5.578 -5.589 -5.678 -5.873 -5.978 -6.138

-4.222

E A 2 11

-9.063 -6.848 -11.025 -12.692

-4.561 -4.612 -4.871 -4.876 -4.954 -5.094 -5.160 -5.268

-4.055

3

X E A 2 11

0.52 0.68 0.60 0.69

0.22 0.22 0.50 0.51 0.51 0.57 0.50 0.58

0.27

R(u)

c o e f f i c i e n t s from T.H. Dunning J r . , and P.J. Hay, i n "Methods of Structure Theory", e d i t e d by H.F.Schaefer I I I , Plenum, New York 1977.

c o e f f i c i e n t s from T.H. Dunning J r . , J . Chem. Phys. 53 (1970) 2823.

millihartree

(a) c o n t r a c t i o n

in

-211.140 -170.604 -226.080 -244.691

[10s6p/5s]

[10s6p/5s]

-0.778 -0.814 -0.732 -0.734 -0.684 -0.627 -0.740 -0.664

-1.274 -1.285 -3.218 -3.232 -3.271 -3.673 -3.379 -3.953

-123.976 -125.082 -139.850 -140.049 -141.182 -145.888 -144.324 -147.882

(3s2p/2s) (4s2p/2s) (4s3p/2s) (5s3p/2s) (5s3p/3s) (9s5p/4s) (5s3p/3s) (5s4p/3s) (5s4p/3s) + 1d(0) + 1p(H) + 1d(0) 1p(H) + 2d(0) 2p(H)

b

-0.489

-1.286

-111.836

a

E

Contracted (3s2p/2s)

i*T

Primitive [9s5p/4s]

E

E 2

Components o f the c o r r e l a t i o n energy o f the ground state of the water molecule using Gaussian b a s i s sets of d i f f e r e n t q u a l i t y *

Basis s e t

Table V I I I :


3

H

H W

C

8

c W

O

1.

GUEST AND

1s : 2p : 3d :


Vector Processors in Quantum Chemistry

WILSON

a = a = a =

0.5 1.0 1.5

3 = 3 = 3 =

31

N = 9 N = 6 N = 3

1.55 1.60 1.65

c a l c u l a t i o n s using exponential basis functions have been performed on the nitrogen, carbon monoxide and boron f l u o r i d e molecules (47) i n c l u d i n g c o r r e l a t i o n e f f e c t s by taking the many-body p e r t u r b a t i o n expansion through t h i r d - o r d e r . Approximately 80% of the e m p i r i c a l c o r r e l a t i o n energy was recovered at the e q u i l i b r i u m nuclear geomet r y of these three species. The c a l c u l a t i o n s were performed on the IBM 370/165 computer at the Daresbury Laboratory and required a considerable amount of CPU time. C l e a r l y , such c a l c u l a t i o n could be made within a few minutes on a vector processor and would, therefore, become r o u t i n e . In 1963, Schwartz (48) emphasised the need to have a systemat i c scheme for extending basis sets. Ruedenberg and h i s co-workers (49-50) have r e c e n t l y r e i t e r a t e d t h i s viewpoint. They developed a technique for s y s t e m a t i c a l l y extending basis sets of the even-tempered type. In t h i s scheme a and 3 are regarded as functions of the number of basis functions, N. This i s necessary i f the b a s i s set i s to be capable, i n p r i n c i p l e , of approaching a complete set. S p e c i f i c a l l y , Ruedenberg et a l (49-50) proposed the e m p i r i c a l forms £n£n3 = b£nN + b* and tna

= a£n(3 - 1) +

a

1

1

where a, a*, b and b are constants. Ruedenberg et a l i n v e s t i g a t e d t h i s approach for a number of atoms within the Hartree-Fock model. The c a l c u l a t e d energies were found to behave smoothly as a funct i o n of the number of basis f u n c t i o n s . I t was shown that the Hart r e e e x t r a p o l a t i o n procedure can provide an empirical upper bound to the basis set l i m i t . Empirical lower bounds to the basis set l i m i t can also be obtained. I t i s important to employ such a systematic approach i n accurate studies so that i t i s possible to assess the convergence p r o p e r t i e s with respect to the basis set. Furthermore, as the basis set i s extended, l i n e a r dependence amongst the basis functions increases. However, i f a systematic approach i s adopted extrapolation procedures can be employed with c o n f i dence to obtain the basis set l i m i t . This procedure has been shown to be u s e f u l i n c a l c u l a t i o n s i n which e l e c t r o n c o r r e l a t i o n e f f e c t s are accounted for (5V). The use of a systematic sequence of basis sets can, of course, be u s e f u l l y combined with the use of a u n i v e r s a l b a s i s set(52). In Figure 8, we d i s p l a y the r e s u l t s of Hartree-Fock c a l c u l a t i o n s on the r a d i a l b e r y l l i u m - l i k e ions L i " , B , C , N , 0** , F , N e , using the basis set given by Schmidt and Ruedenberg(56) s p e c i f i i c a l l y for the b e r y l l i u m atom. I t can be seen that for the p o s i t i v e ions the b a s i s sets give a uniform convergence r a t e . In +

2+

3+

+

5 +


6+


Figure 8.

Plot of ln(E[n"s] — Bin's]) against basis set size for beryllium-like ions.


3

GO H

s

m

X

2 ο

GO

H W

d

§

w

Ν)

1.

GUEST AND WILSON


Chemistry

33

Figure 9, we show the c a l c u l a t e d c o r r e l a t i o n energies f o r these systems. Again with the exception of L i " , uniform convergence i s observed. I t i s envisaged that by employing a u n i v e r s a l systematic sequence of even-tempered b a s i s sets on molecules such as N / CO and BF, we can employ e x t r a p o l a t i o n procedures with some confidence and i n v e s t i g a t e b a s i s set l i m i t s . This w i l l enable quantum chemic a l c a l c u l a t i o n s to approach a chemical accuracy of 1 m i l l i h a r t r e e for small molecules. 2


Concluding Remarks I t i s c l e a r that the new generation of computers - supercomputers when used i n conjunction with t h e o r e t i c a l developments such as the l i n k e d diagram theory or the u n i t a r y group theory, i s going to have a s i g n i f i c a n t impact on the accuracy a t t a i n a b l e i n quantum chemistry and on the s i z e of problem which may be t r e a t e d . The implementation of various standard quantum chemistry packages on the CRAY-1 computer at Daresbury was r e l a t i v e l y s t r a i g h t forward, but s i g n i f i c a n t recoding was necessary to take advantage of the vector processing features of t h i s machine. The f o l l o w i n g points emerged from t h i s recoding. (1) In the CRAY-1 a Gaussian i n t e g r a l s program may be d r i v e n at 35 Mflops, a large c l o s e d s h e l l SCF with a sparse l i s t of i n t e g r a l s or Supermatrix at 10 Mflops, while a smaller SCF with a non-sparse Supermatrix may be driven at 135 Mflops. Procedures such as SCEP may be performed at 135 Mflops, w h i l s t a four-index transformation of the 2-electron i n t e g r a l s w i l l also proceed at 135 Mflops i n large cases, i n d i c a t i n g the CRAY to be between 5 and 25 CDC 7600 i n power when given a p p r o p r i ate code. (2) To achieve the above performance f i g u r e s some programs r e q u i r e large amounts of main memory to be a v a i l a b l e . (3) Some a p p l i c a t i o n s (e.g. Fock matrix construction) w i l l demand very high t r a n s f e r r a t e s from backing store i f they are not to become severely I/O bound. (4) A requirement f o r s c a t t e r / g a t h e r hardware and other aids to sparse matrix handling has been i n d i c a t e d , s p e c i f i c a l l y i n the case of SCF c a l c u l a t i o n s and i n the s o r t phase of the 4-index transformation. The requirement w i l l , we suspect, be even more severe i n the case of C o n f i g u r a t i o n I n t e r a c t i o n c a l c u l a t i o n s . (5) The CRAY FORTRAN compiler, CFT, sometimes produces n e a r l y opt i m a l code (as f o r example i n Gaussian i n t e g r a l s e v a l u a t i o n ) ; sometimes an increase i n speed by a f a c t o r of 3 i s observed on proceeding to the assembler, CAL. T h i s f a c t o r of 3 i s o f t e n present i n rather simple a p p l i c a t i o n s such as matrix m u l t i p l y . In our work we have been unable to o b t a i n more than 50 Mflops computation rate from the CFT compiler, and t h i s r e f l e c t s a lack of r i c h n e s s i n our codes. The e v a l u a t i o n of a high order polynomial




34

6s

1 8s

1 10s

1 12s

1 14s

1 16s

1 18s

1

20s

Basis set Figure 9. Plot of the correlation energy, given by the [2/1] Padi approximant to the third-order energy, against basis set size for beryllium-like ions.


1.

GUEST AND WILSON

R

±


= ((((X +a)*X +b)*X +c)*X +d) i

i

i

i

Chemistry

(a,b,c are

35

constants)


i s an example of a r i c h code ( f o r each f e t c h of X and store of R a great deal of F l o a t i n g p o i n t a r i t h m e t i c o c c u r s ) , and here the CFT compiler would produce code of 135 Mflops given a polynomial o f s u f f i c i e n t l y high order. The new machines have enabled new areas of research to be i n v e s t i g a t e d . We have described some of our work i n the c o n t r i b u t i o n of higher-order e x c i t a t i o n energies to c o r r e l a t i o n energies using diagrammatic many-body p e r t u r b a t i o n theory and our work aimed at the development of the large b a s i s sets which can now be employed i n molecular c a l c u l a t i o n s together with schemes f o r systema t i c a l l y extending them. Acknowledgements We wish to express our g r a t i t u d e to Dr. V.R. Saunders f o r h i s invaluable a s s i s t a n c e , without which t h i s work would not have been possible.

Abstract The impact which vector p r o c e s s i n g computers are having on ab initio quantum chemical c a l c u l a t i o n s will be considered, g i v i n g s p e c i a l emphasis to the experiences of United Kingdom s c i e n t i s t s . Recent work using the CRAY-1 computer at the S.E.R.C. Daresbury Laboratory will be d e s c r i b e d . The CRAY-1 computer i s a t the cent r e of a network p r o v i d i n g large s c a l e computational facilities f o r u n i v e r s i t i e s i n the U.K. Experience of running and subsequent v e c t o r i s a t i o n of c e r t a i n standard ab initio packages will be d i s cussed. I n t e g r a l e v a l u a t i o n , s e l f - c o n s i s t e n t - f i e l d and o r b i t a l transformation phases of quantum chemical studies will be c o n s i dered together with both the many-body p e r t u r b a t i o n theory and c o n f i g u r a t i o n mixing approaches to the e l e c t r o n c o r r e l a t i o n problem i n molecules. Not only are improvements to the t r a d i t i o n a l algorithms which are made p o s s i b l e by the use of vector processors discussed, but a l s o new areas of research which such comput e r s open up are o u t l i n e d . Future developments are considered briefly.


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The Use of Vector Processors in Quantum Chemistry - ACS Publications

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