https://ntrs.nasa.gov/search.jsp?R=19660006957 2020-06-04T19:05:03+00:00Z

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TIME-DEPENDENT WPERVIRIAL THEOREMS AND THE LIKE

FOR VARIATIONAL WAVE FUNCTIONS * by

Saul T. Epstein

Theoretical Chemistry Institute and Physics Department

University of Wisconsin, Madison, Wisconsin

ABSTRACT

Conditions are given under which optimal variational

wave functions will satisfy time-dependent hypervirial

theorems, Hellmann-Feynman theorems, etc.

* This research was supported by the following grant: National Aeronautics and Space Administration Grant NsG-275-62.

Let *' and 5% be optimal time dependent variation wave and H respectively.

Hx Y functions appropriate to the Hamiltonians 1 9% and @I thus satisfy the variational equations

Let us suppose that

with 1 a small parameter, is a possible variation of

From (lb) this then implies

9, .

1

2

S i m i l a r l y i f

wi th F+ t he Hermitian conjugates of F , i s a poss ib l e v a r i a t i o n

of % then (2a) implies

Sub t rac t ing ( 6 ) from ( 4 ) and us ing t h e Hermitian proper ty

of Hx w e then have

or

or f i n a l l y

c

For HXz H ( 7 ) might wel l be c a l l e d a time-dependent (off- Y

d iagonal i f 9% jk 9% ) h y p e r v i r i a l theorem while f o r F = 1,

( 7 ) becomes the time-dependent i n t e g r a l Hellmann-Feynman theorem

of Hayes and Par r , though now f o r opt imal v a r i a t i o n a l func t ions . 2

Now l e t us suppose t h a t

yea 1 wi th x a/parameter, i s a p o s s i b l e v a r i a t i o n of ex . Then

from ( l a ) we have

while from ( l b ) we have

A l s o l e t us suppose t h a t

i s a poss ib l e v a r i a t i o n of %x so t h a t from ( l a ) we have

c

c

4

Consider now

Using (9) and (10) one r e a d i l y f i n d s t h a t t he l a s t two terms

on t h e r i g h t hand s i d e can be w r i t t e n as

while from ( l l ) , t h e l e f t hand s i d e of ( 1 2 ) can be w r i t t e n as

P u t t i n g a l l t h i s together then, (12) can be w r i t t e n

or

which i s the time-dependent d i f ferent ia l Hellmann-Feynman theorem

of Hayes and Parr , though now for optimal variational functions. 2

5

Footnotes and References

1. J . Frenkel, Wave Mechanics, Advanced General Theory

(Clarendon Press, Oxford, England, 1934) p . 436.

2 . E . F . Hayes and R . G . Parr, J . Chem. Phys., 43, 1831 (1965).

They a l s o derive equation ( 7 ) for exact wave functions.