Determination of density matrix components for multiconfiguration wave functions and Hamiltonian interaction matrice

Harriman, J. E.

Welsh 

NASA Technical Reports Server

A Dens i ty  M a t r i x  Approach t o  
* ,f lv iu l t iconf igura t  i o n  Calcu-la"; L LOnS 
'b Y 
John E .  'riarriman 
T h e o r e t i c a l  Chemistry I n s t i t u t e  
Univers i ' ty  o f  Wisconsin 
Madison, Wiscczisin 53706 
ABSTRACT 
D e n s i t y  ina t r ix  c m p o n e n t s  a r e  de te rmined  f o r  a wave f u n c t i o n  
o f  mul t i conf igu l r a t io i l  t y p e ,  The a r r a y  c o n t a i n i n g  i n f o r m a t i o n  
r e q u i r e d  t o  o b t a i n  t h e  two-e l ec t ron  chaiige d e n s i t y  n a t r i x  can  
a l s o  b e  used t o  c o n s t r u c t  t he  N-e lec t ron  ( s p i n - f r e e )  harn l l ton ian  
c o n f i g u r a t i o n  i n t e r a c t i o n  m a t r i x  f rom t h e  two-e l ec t ron  reduced  
h a m i l t o n i a n  m a t r i x .  T h i s  a r r a y  can  be o b t a i n e d  i n  s e v e r a l  ways 
and s u g g e s t s  a usefu l .  c o n c e p t u a l  o r g a n i z a t i o n .  
& 
Research  suppor t ed  by t h e  N a t i o n a l  Aeronab t i c s  and Space 
Adrn in i s t r a t ion  Gran t  NGL 50-002-001 and t h e  N a t i o n a l  S c i e n c e  
Foundat ion  Grant  GP- 12832. 
4- Paper  g iven  a t  t h e  I n t e r n a t i o n a l  SyxposLum on Atomic,  Molecu la r  
and S o l i d  S t a t e  Theory and Quantum Biology a t  S a n i b e l  I s l a n d ,  
F l o r i d a ,  J a n u a r y ,  1970. 
https://ntrs.nasa.gov/search.jsp?R=19710009662 2020-03-12T00:09:24+00:00Z
good way of  i n t e r p r e t  iqg many-e l e c r r c n  wavefuactio~s a 
a specif ic  wavefunct ion  f o r  a p s r c i c t i l a r  sys tem or a gene ra l  
wavefunct ion of' seaie t y p e .  The d e n s i t y  m a t r i x  (DN) i s  perriculsrly 
u s e f u l  2 3  C C E ~ ? : ~ K ~ . J ~  f u n c t i o i s  of . i<fPere2t  f ,ypes-  'it: m y  be 
p s s l h i c ?  on xhe '5as-i.s of ' a  dens-5r.y niar-r.fx m a i y s i s  c c  rriake r a t h e r  
genera l  s ta terr tents  about the S c s c r i p t i o n  sny  wavefunct lon  of a 
p a r t i c u l a r  r y p e  can m k e ,  as in t h e  ca5e of a spln-projected 
d i f f e r e n t  Grbj tals fcr- d",fr'erenrr sp ins  sir& 3eterminant: .  
T h i s  may be 
4 , 5  
The.densit.-y maLrlx foren i la t ion  may a l s o  facilitate C B I C U L B C ~ O T I  
by sugges t ing  an organjzacion whi-ch i s  no t  apyJarenr. from t h e  
w a v e f m c t i o i  i t s e l f ,  Ea f a c t ,  4 t  would be desfrable t o  work 
with t h e  d e n s i t y  matr-fx d i r e c t l y ,  w i t h o u t  i n v o l v i n g  an N - e l e c t r o n  
wave f u n c t i o n ,  To do so  would r e q u i r e  a s o l u t f a n  of t h c  
4 3 - r e p r e s e n t a b L i 5 t y  probleni, a g o a l  which has  t h u s  f a r  rernsirled 
e l l u s i v e .  One p o s a t b i l i t y  is t3 c o t l s t r u c t  an N-repressntabie 
d e n s i t y  m a t r i x ,  contafning v a r i a t i o n a l  paramerers from a wdveEiinction 
of sone t y p e ,  and t o  wsrk w i r h  ice The func t ior i  i s  of course 
FmplicLtZy d e t e r m i n e d ,  b u t  need not a p p e a r  e x p l i c i t l y  a t  any tirne. 
Exper ience  w i t h  two-e lecr ron  c a l c u l a t i o n s  shows t h a r  i t  i s  
d e s i r a b l e  t o  i n c l u d e  
One would rhus  wish t o  use  r -contairlling geminals in expanding 
t h e  2-mat r ix .  Obv:cuslp, however, o n l y  a finite number of g e d n a l s  
will be used and t h u s  the rank of t h e  2-matrix w i l l  be f i n i t e .  It 
52 e x ; > l i c i t l y  i n  a varistional f u n c t i o n .  
12 
2 
can be proven 
w i t h  no knova 
9 f o r  a l j  odd N > i  and i s  co;.jccttxl f o r  C V E ~  N > 2 ,  
c o u n t e r  examples bu t  as y e t  no p r o o f ,  t h a [  the  rank 
CL t r e a t m e z t  i s  s t rd ighzfor -ward ,  b u t  v e r y  many terms a r e  r e q u i r e d  
t o  g ive  gocd r e s u l r s ,  It t s  haped t h a t  hy opfimizL::g o r i 3 i t a l s  a s  
w i l l  a r e l a r r i v e l y  small niinber of c o n f i g u r s t i m s  w i l l  s u f f i c e .  
Densitly MaLrlx Ccnponents 
The one- an9 rwo-e lec t ron  r e d i i c d  d e n s i t y  nnitr ices are  g iven  
2 
i n  t h e  cbwden n o r n a l i z a L i o n  by 
where a s  u s u a l  x stands f o r  t h e  s?ace and s p i n  c o o r d i n a t e s  cE 
e l e c t r o n  i .  These d e n s i t y  m a t r i c e s  can be r e s o l v e d  i n t o  spi.n 
i 
3 
The change * d e n s i t y  xacrix 
-t Y"' y o  = p-+ 
and t h e  s p i g  r i e n s i t y  n,atrix 
yz  = yf'- - y - -  ~ 
which v a n l s h e s  i d e a t i c a l l y  i f  M = (1. 
t e n s o r a l  curnpozents of y, 
are the  i r r e d l l c i b l e  $ ? E n -  s 
14 
A sim113r r e s o l u t i o n  of r i s  p o s s j - b i e ,  
( 6 )  
The only  ccmponent w i t h  which w e  need be p a r t j e u l a r l y  concerned  
is t h e  two-e l ec t ron  change d ~ n s i t y  matrix 
+- r+-+-  r+*+ + I-. 0 (7)  
I* & m Y  - + + - c +  r Q  = r 
4 
The i n t e g r a l  ope ra t c r  of which 1"is t h e  2.;er1ia1 a c t s  on rwo-e lec t ron  
will have zero m a t r i x  eiements between symne t r i c  a.13 a r i t i  s y r m e t r i c  
1 7 
geminals  a 
a r e  t h e n  formed a s  
T h i s  decomposi t icn  i s  of cmrse  independent  of t h e  p a r t i c u l a r  
b a s i s  chosen. 
If t h e  hamiltonian for t h e  N-electron sysce- is made rip of 
one- and two-electron terms 
5 
Then a reduced harn i l ton lan  11 can be  de f in2u  a s  r 
4 - - [ + ' . f i  'T f ( 2 1 - J  4- g H 2 :  
I-Ir - N - ?  
wi th  t h e  p r o p e r t y  t h a t  t h e  energy  of  t h e  sys tem i s  g iven  by 
I f  H and rhus  H a r e  s p i n  f r e e ,  I' can  be r ep lhced  by r o *  The 
reduced hamil ro-.iia.n i s  a symmetric t w o - y a r t f c i e  o p e r a t o r  corrirnuting 
1" 
w i t h  P12 a l s o ,  and t h u s  
Th i s  s e p a r a t i o n  of  E i n t o  c o n t r i b u t i o n s  from s y r m e t r i c .  and 
an t i symmet r i c  components i s  c l e a r l y  r e l a t e d  t o  D. id. S m i t h ' s  
s e p a r a t i o n  i n t o  E "  and E I 18 I n  t h e  p r e s e n t  c a s e ,  it i s  c l e a r  
t h a t t h e  an t i s yme t r i c gemir; a 1 s invo l v e  a s yxime t r y - i n  d u c ed 
c o r r e l a t i o n - l i k e  e f f e c t  co r re spond ing  t o  t h e  Fermi  h o l e ,  w h i l e  no 
such e f f e c t  i s  e s s e n t i a l l y  p r e s e n t  f o r  t h e  symmetric geminals .  I n  
f a c t  t h e  two d e c o r q o s i t i o n s  are  r e l a t e d  by 
I f  one i s  concerned w i t h  coir;l3utational a s p e c t s ,  f u r t h e r  
S a 
advantage  can be t aken  of t h e  f a c t  that:  
a r e  symmetric m a t r i c e s .  
and , as well a s  H 
--T 
6 
(In g e n e r a i  they  a r e  he r tn i t<an ,  b u t  i n  prdccice t h e y  will normal ly  be  
r e a l )  
X 
I H X  - 
i j , k l  “ki i j 
The a r r a y s  an6 t r a c e  o p e r a t i o n  can a i s o  be l i n e a r i z e d  “Co g i v e  e x p r e s s i o n s  
o f  t h e  form. 
E = l H T .  
1.1 P b  
T h i s  p r o c e s s  i s  s t r a i g h t f o r w a r d  and welHCnown, s o  w e  w i l l .  n o t  g i v e  
d e t a i l s  h e r e .  
MulticonfiguratienFunctions. 
A m u l t f c o n f i g u r a t i o n  (MC) wavefunct ion  of t h e  type b e i n g  con- 
s i d e r e d  h e r e  can be w r i t t e n  a s  
(17)  
where t h e  s i n g l e  index  i s t a n d s  f o r  a s e t  of  i n d i c e s  i + ( K L j ) ,  and 
Q i  = A T K L e j K  
i s  t h e  an t i symmetr ized  p roduc t  of a s p a t i a l  p a r t  T 
p a r t .  We w i l l  n o t  c o n s i d e r  e x p l i c i t l y  arly symmetr ies  o t h e r  
,-and a s p i n  
than  s p i n  and p e r m u t a t i o ? a l ,  b u t  of c o u r s e  o r b i t a l  a n g u l a r  momentum 
o r  molecu la r  p o i n t  group sy rme t ry  C ~ R  be t r e a t e d ,  
The s p a t i a l  f u n c t i o n  i n v o l v e s  K. dou3l.y occupied o r b i t a l s  w i th  
k d  and X si :ngly occupied  o r b i t a l s  w i t h  i n d i c e s  5 i n d i c e s  k 
Z 4  * ,  . Z A  c l e a r l y  2 K  + X = N e The i n d i c e s  K and i 
7 
The s p i n  f u n c . ~ i o n  i s  theri t aken  a s  
; X.) 
j 
where 8 t h e  j u h  s p i n  e i g e n f u n c t i o n  for  rrhe l a s t  k e i e c t r o n s  
w i t h  t h e  d e s i r e d  S and M v a l u e s .  
i s  then  a s p i n  e i g e n f u n c t i o n  w i t h  t h e  s a m  quantum numbers* The 
Thc a n t i s y r m e t r i z e d  producr: Q i  
S 
r e s u l t s  a r e  e s s e n t i a l l y  i .ndependent o f  how t h e  s p i n  e i g e n f u n c t i o n s  
a r e  chosen ,  so lolzg a s  a c o n p l e r e  set i s  u s e d ,  It i s  conven ien t  
f o r  some purposes  i f  they  a r e  chosen or thonormal ,  
A l a r g e  number o f  d i f f e r e n t  t y p e s  o f  wave f u n c t i o n s  can be 
w r i t t e n  i n  t h i s  form. C l e a r l y  inc luded  a r e  t h e  c o n v e n t i o n a l  CI 
f u n c t i o n s ,  i n  which t h e  o r b i t a l s  are  f i x e d  and on ly  t h e  c o e f f i c i e n t s  
a r e  opt imized  i n  a v a r i a t i o n a l  c a l c u l a t i o n  and Oc.3C f u n c t i o n s ,  'i 
i n  which b o t h  c o e f f i c i e n t s  and o r b i t a l s  a r e  opn imized .  It i s  a l s o  
o f  i n t e r e s t  t h a t  a v a r i e t y  of "Extended H a r t r e e  Fock" f u n c t i o n s  
can be  expres sed  i n  t h i s  form by t r a n s f o r m a t i o n  t o  an a p p r o p r i a t e  
set of  o r t h o g o n a l i z e d  o r b i t a l s ,  such  a s  t h e  e i g e n f u a c t i o n s  o f  t h e  
cha rge  d e n s i t y  ma t r ix  y a F o r  s p i n - p o l a r i z e d  HF, s p i n  p r o j e c t e d  0 4  
o r  spi.n ex tended  HF, and even  sp in -op t imized  HF f u n c t i o n s ,  t h e  
r a n k  o f  y o  does  nor. exceed N, and t h u s  t h e  wave f u n c t i o n  can b e  
expanded i i i  c o o f j g u r a t f o n s  i n v o l v i n g  n o t  more than  N or thonormal  
o r b i t a l s .  A C I  f u n c t i o n  i n  wh ich  a l l  s i n g l y  e x c i t e d  c o n f i g u r a t i o n s  19 
8 
o r  
x = s o r a  
y = 0 o r  2 ,  
These equarioas can be take3 zc Q____ d e f i n e t h e  arrays f, py. 
With t h e  reintroducticn of U li? p l a c e  of  i j 9 k !€or convenience,  
t h e  e n e r g y  c a n  now be writKen 
9 
h t e q r e t s t i o n  a r i s e s  i f  t h e  s u n  oke r  IJ 4s done firc,,t t c s  g i v e  
* 
c I  c c E =  T U  t !I 
(2;) ' 
Eqn  ( 2 4 )  is  the u s u a l  CI enefgy e x p r e s s i o n  and - €IC' is t h e  m a t r i x  
of t h e  (?~-e iec t ron3-hami l ton ian  i a  t h z  b a s i s  of the c o n f i g u r a t i o n s  
@ .  , The a r r a y  - P t h u s  a l so  p r c v i d e s  a way of c o n s t r u c t i n g  t h e  CT 
m a t r i x  from t h e  e'Lemezt; of t h e  reduced ha rn l l t on ian  ma t r ix .  
I 
The i n f o r n a r i c n  which i s  c o n t a i n e d  i n  the a r r a y  P is i m p l i c i t l y  - 
present:  i n  any CI o r  PIC c s l c u l a t i o n .  I f  each c o s f i g u r a t i o n  i s  
taken s imply  as  an an t i symrre t r i zed  p roduc t  of s p i n  o r b i t a l s  w i t h  
no r e s t r l c c l o n  t o  produce S ' e i g a - ~ f u n c t l o n s ,  t hen  - P i s  p u r e l y  
cornSlns tor ia1 .  It el&.;nencs a r e  0, kll, 2 ani? a re  comple t e ly  
de te rmined  once a numbering scheme 2s dec ided  f o r  t h e  c o n f i g u r a t i o n s .  
I f  the f u n c t t o n  i s  a CI involv:'.ng only  doubly occupied  o r b F t a P s 3  
e s s e n t i a l l y  t h e  same i s  t r u e .  Fo r  func t l ions  i n v o l v i n g  non- t r i v i a l  
s p i n  c o u p l i n g ,  t5e d e t e r m f z a t i o n  o f  i s  more d i f f i c u l t .  
PO 
Partlcular elemencs 7.7: 1; have diii-erenrr va lues  depending on how t h e  
We have seer1 t h a c  t c  does  no t  rrratter, e x c e p t  pcssfblyfrorn a 
s t a n d p o i n t  of co:ivenience, j f non-or thogonal  s F i n  eigenflrnc t ions 
a r e  tised. It: has  beea  assumed th roughou t ,  however,  t h a ~  the 
o r b i t a l s  a r e  orthCgi)nsl .  The use of non-cr thogousl  o r b i c a l s  
e s s e n t E a i l y  Tnvaltdaces t h e  q p r o a c h  b e i n g  p r e s e n t e d  h e r e :  s i n c e  
then  depends or? orb i t a l .  o v e r l a p s  and w i l l  change a s  cfie o r b i t a l  s e t  
i s  t r ans fo rmed ,  
Vhen e l emen t s  of t h e  CX m a t r i x  have bee3  construel~d i n  
p r a c t i c e ,  i t  hds n o t  u s u a l l y  been i n  te rms  of a reduced ha in i l ron ian  
marrix.  Ti I s  of  f n t e r e s g  EO n o t e ,  however, r h a t  t h e  nugber  o f  
p o t e n t i a l l y  i n d e p e i d e n t  elemerlts i-.1 H -
number of  p o t e n t i a l  l y  Iadependent coulomb p l u s  exchange i n t e g r a l s  
i n   he s ~ i n e  b a s i s .  ;2 d e f l g i n g  the red?iced hamilronian a l l  t h e  
o n e - e l e c t r o n  i z t e g r a ' i s  cazi be absorbed  and need no t  be c o n s i d e r e d  
e x p l i c i t l y .  
S a 
p l u s  2 i s  rhe  sme as  t h e  
11 
Thr;s f a r  oxlly per:ndtt5timiil and s7ir.l syirmetry tlave Seen 
2 
c o n s i d e r e d .  Other  symmetries, L behav io r  f o r  an atcm o r  p o i n t  
group symmetry f o r  a molecule ,  a r e  a l s o  of i n t e r e s t .  The con- 
f4gurac5ons can be l i m i t e d ,  UI’ s i n g l e  o r b i t a l  p r o d - . ~ t s  replaced 
by s m a l l ,  €‘%?:e3 l i c e a r  cwiSi.nati  C)IS of pro6ucts  t o  give t.he 
d e s i r e d  behsvi .or .  There w i  1 f be  a3diti.r;:ial synrrrieirry f a c e o r i n g s  of 
H ,  and y o  e 
c o n s t r u c t i o n s  of P a  
These f e a t c r e s  cz.1 al;o be dea l t :  with by a p p r o p r i a t e  
D i s c u s  sL02 
A s  h a s  been p o i n t e d  o u t  above, t h e  inforniati.on i l eces sa ry  t o  
c o n s t r u c t  t h e  a r r a y  P e x i s t s  imp?icftly i n  a n y  PIC o r  C f  c s l c u l a t i o n a l  
scheme, The novel f e a t u r e  of che p r e s e n t  zpproach i s  To s e p a r a t e  
and cons ide r  explicitly v a r i o u s  a spec t s  of t h e s e  apprc.aches, Such 
a s e p a r a t i o n  i s  v e r y  u s e f u l  c o n c e p t u a l l y ,  and may be of some a i d  
i n  computa t ion .  
The f u n d a x e n t a l  e x p r e s s i o n ,  on which d i s c u s s i o n  can he based ,  
i s  Eqn (23 ) .  I n  t h i s  e q u a t i o n  t h e  t h r e e  a s p e c t s  of t h e  MC 
approach a r e  a p p r e n t :  
1) The dependence of t h e  energy  on t h e  expsuston 
c o e f f i c i e n t s  C as  i n  any CI o r  MC e x p r e s s i o n .  iY 
2) The dependence on t h e  o r b i t a l s  i n v o l v e d ,  which 
a;>pears he re  i n  t h e  reduced ha rn i l t on ian  matr ix .  Any 
t ray is format ion  of o r b i t a l s  w i l l  induce  a t r a n s f o r m a t i o n  of 
t h i s  m a t r i x .  
1% 
3) The c o x h j n a t c r i a l  and synimetry s s p e c t s  which are  
contaLi;ed in  he cotipPL~g a r r a y  7, 
The .;jlcter;ni.:Iction of 2 x i 5  K€x)s~~? relacionships w h i c h  depend 
oil i t  can be  done inGepexje..?tiy of the "dyrlsmics" cif il particular v a r -  
i a t l o n a l l y  optimized f i ; i i ~ : t i o ~ .  IC i g  t h u s  m r e  convenient  t o  
exax !.;I e s e p 2 r at e 1 y a?i d exy: 1 .i c i. t l y t h e  s t YILC t lir 9 1 a s 2 e c. t. s of the  
problem. I n  p a r L i c u i a r ,  the e x a n t i r 1 a t . f ~ ~  o f  . these aspec ts  in 
densi ' iy  mat r . ix  cerms r ep i - e se i~~s  a g r e a r  s l r n i p l i f i c a c l o n  o v e r  
i m p  1 i c i t c x i  s i. d e r a f i o  it o f c oriry L i. c a t e d MC wme furl c :: ion  s 
I
-.. 
T h i s  approach a l s o  r e p r e s e n t s  x h a t  might  be called an avoidance  
( r a t h e r  t h a n  a s o i u t i o a )  ~f the N - r e p r e s e n t a b i l i t y  problem. By 
for lnal ly  c o n s t r G c t i n g  a clej3sj.ty m a t t i x  from a wave funcrior? ,  
N - r e p r e s e n t a 3 i l Z t y  i s  a s s u r e d ,  C e r t a i n  features w1iFc.h w i l l  be 
present-  i f  che N- re? resen tab i l i cy  p r o b l e n  c a n  be soi.ved t h m  
become a p p a r e n t .  I t  i s  s t i l l  n e c e s s a r y  t o  d o  a v a r i a t i o n a l  
c a l c u l a t i c n .  32e k i n d  of v a r i a t i o n  is t h e  opti .on.i .7ation of the 
o r b i t a l  s e t .  3ne can, f g r  example, ps rarne t r ize  t h e  orthogcmal  
t r a n s f o r m a t i o n  from a fundamental  b a s i s  s e t  t o  t h e  o r b i t a l  s e t  
i n  terms cf t h e  e x p o n e n t i a l  of 8:i a n t i s y m m e t r i c  mstri.x. The best  
ways of d e a l i n g  x i t h  t h i s  t r a n s f o r m a t i o n ,  p a r t i c u l a r l y  when t h e  
s i z e  of the  b a s i s  se t  i s  inuch l a r g e r  ir'han the niimber of o r b i t a l s  
a c t u a l l y  x e d ,  i s  a d i f f i c u l t  one and n e e d s  much more a t t e n t i o n .  
This nee5  becones a p p a r e n t  i n  t h e  p r e s e n t  f o r m u l a t i o n .  
13 
Noteo 2nd hc:fere;lces 
1, :: i s  c i e a r l y  5eyond rhe scoae  of r h c  p r e s e n t  paper  LG p r o v i d e  
--._II-y-.1.1-9--.-- --.- 
a su rvey  of t h e  d m s i r y  m a t r i x  i f t e r s t u r e .  E a r l y  work w i t h  
Madison, 1969) e 
2. P ,  0 ,  L‘&wclin, TIiys. Rev. 97, 1474 (1955) 
3 ,  R. McWeeny, Rev, Mod. Phys. 33, 335 (1960) 
4 ,  J. E .  Harriman, J. Chem, Phys.  52, 2827 (1964) 
5. A ,  Hard i s son  and 3. E ,  Ilarr:’man, J. Chern, Phys. 4 6 ,  3639 (1967) 
6. K ,  M. Sando and J. E .  Harr iman,  J. Chem.  Phys .  
- 
7 ,  J, E.  Harriman and R. M. Sando ,  J .  Chem, Phys.  - 48,  5133 (1968) 
8. A. J .  CGlexlan, Rev. Mod. Phys.  - 35, 668 (1963) .  See  a l s o  t h e  
r e f e r e n c e s  g i v e n  i n  n o t e  1. 
M, E ,  Rri.sk.ai, Phys.  Revo -. 183,  129 (1969) .  
10. An example of  t h e  u s e  of a p p r o x i a a t e l y  N - r e p r e s e n t a b l e  d e n s i t y  
matrices may be found i n  t h e  work of Garrod and Ros ina ,  s i n c e  
they impose n e c e s s a r y  b u t  n o t  s u f f i c i e n t  c o n d i t i o n s  f o r  N -  
r e p r e s e n t a b i l i t y  and proceed w i t h  v a r i a t i o n a l  c a i c u l a t i o n s .  
11. J. Si rnms  and J ,  E .  Harrirnan, t o  be p u b l i s h e d .  D. W.  S m i t h ,  
t o  be pr ;bl ished.  
2 2  0 
13 n 
s i  
i 4  
15, 
16 c 
2'7 0 
18. 
19  a 
2 0 ,  


A density matrix approach to multiconfiguration calculations

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server