- Email: [email protected]
Variational bounds to the overlap
tolume
3 1, nukber
CHEMICAL
2
PHYSlC,S LETTERS
1 hkuch 1975
‘,
VARIATIONAL BOUNDS TO TIIE OVERLAP
”
Thomas HOFFMANN-OSTENHOF and Maria HOFFMA&%-OSTENHOF lnstihct fir Strchlencllemie Received 29 October
im Max-Flat@-Institur
fiir Kohlenforschung,
D-4330 hfiilheimjRuhr.
Germnjzy
1974
Starting from a closed expression
for the overlap, variational upper and lower bounds to the overlap ae derived by means
of operator inequalities.
1. Introduction
2.
The evaluation of upper and lower bounds to quantum mechanical properties [l] usually required the estimation of the overlap Sk =(~I~k);(~l~O)=(~kl~Clk) = 1; 50is an approximation to a solution ak of the lime independent Schrcdinger equation HGk = Ek $k (k = 0,1,2,. . .),ff being self-adjoint. niloreover lSk I itself is a measure of the accuracy achieved in an actual calculatioe [1,2]. The importance of variational bounds to the overlap stems from the fact that ISkI can be approximated in a systematic manner and equality can be,achieved at Least in prikiple [2]. These variational bounds may be written in the following way
For every function and lower
bounds
I(% x) 2 ISk 12,
x the function& respectiveIy
J(& x) =q
in (1) yield
upper
A closed expression for the overlap Let us define an operator Igk such that
‘@ = $kf$,
where we assume Sk + 0 and the corresponding eigenvalue Ek is non degenerate. Such an operator Ivk will be called wave operator [S, 51. From the deftition it follows immediately that IQ
= (,~llQu&)-l.
(2)
We can distinguish between two kinds of variational
bounds. (a) The equality sign holds only if x0 t = \clk [2-l] or.(b) x,, 5 Gk. In contrast to boun& of type (b), bounds of type (a) require a x which is a better ap-’ proximation tp Gk than rp, if high accuracy is desired. !n the foilowing we wti be concerned only with bounds : ,, oftype (b).
(4)
We note further that QlW&00) = 1.
(5)
Wave operators are often encountered in partitioning technique and perturbation theory [7]. In general these operators are not self-adjoint. However For the followirg derivations self-adjoint wzwz oyerators will be useful. Let M-k = rn(H-
12.
(3
m f 0
Ek) + ILp>(~I,
(6)
being a co&plax parameter. Obviously
MkGk = Q.
(7)
AssumingAIk non-singular and m real thenJi,-’ represents a self-adjoint wave operator. Substituting rVk= J%$’into (4) we obtain lSJ2
= c~lM~21~r’.
(8)
The problem is now to find b&n& to the expectation value in (4) or (8). Tkwill be done by means of oper‘ator inequalities.’
:
;. .‘,
277
Volume
3!, number’2
3. Upper bouqds
Substituting
to the overlap
..A-Jk3-l To d&iv& a vkiational upper bound stk from.the operator inequa!ity A-’ 2 1x1 c&I
to Isk I2 we. ,.
IXF~~l,
(9)
where_&> 0. x is ti arbitrary function and TX, (x]A Ix>-~’ (xl is called an inner projection A-l .‘Therefore
cf:;d :
i$? > 14 Ix$/(xlA
IX>
(10)
(10)
with ($Wk)-’ and {‘with 9, (4) and uppkr boynd
lead to the following
IS,12 ~~xl~rv,‘rv,~-‘Ix~/lclplX~12. Fo; the special wave operator
(10
Wk =ilI~‘, ‘.
w,=‘[m(H-Ek)+I~)(~(]-1
that is. td say ,’
.,
(9
minimisation of (11) with respect to the complex parameter m-produces ihe well known upper bound [9] (13)
to the overlap
The derivation oflower bounds to ISkI will again be based on an cperator inequaliiy. Let A and B represent self-adjoint operators with A > B > 0.Consider the identity
-/g-l =B”
L (B-1. - A-1).
(14)
If A > B > 0 it can be shown that B-l >A-’ > 0 [7] . Therefore 8-l - Am1 > 0.Hence we cti appl the Y method of inner pr,oj$ions [S] to A_1 - B- with an arbitraj function 77
-B-l,_/' ,, :.
”
‘. +-%4-‘)1;!~~18-1 . . ,‘, ~.?&c
I ‘,
-,.
,-,
”
..
operator.
-
(17)
That means
(~I~__l,,,.,,‘l,-l,,,_~lc~le-‘a~rlX,i~ (18) (y_IAB% -A lx> for ELIarbitrary function t, provided the corresponding integrals exist. Equality is achieved for x0 t =A $. Not& that xopt does not,depend upon B. 1Qe want-l to emphasize that for both variational principles (10) and’ (18) xopt turns out to be identical. Furthermore A-’ does not occur on the right hand side of (18). Inequality (18) will be the starting poini for the ‘derivation of lower bounds to ISk12. Identifying A with an appropriate chqice for the operator B (W,t WJl has to be made in order to apply (18). In the following we shall consider,two s&Al choices for B. i:) First we note that for .properly chosen non-singular IV-!l the operator (W+W )-I is positive definite and thkercfore a p > 0 e&s iuch that -1
>pI>O.
(19)
We put B = pl and E = cp and insert into (18). Taking (4) into account we obtain the variational lower bound to the overlap
I(#p,)-‘-plx)12
ISkI’ >p_ 1 [ For-the
cxl(rv,fru,)-*-p(Co,‘W,)-‘Ix
wave operator
IS&=P
[
1-
led_12 -PlX~121
I~o>cipl~4
_1
1
im(H-~~~f!~~~~l~~lX~l-~ ._ considerkb
. (20) J
(12) with real m this becomes
[lbpl &y)+
x~[cxI{m(H--q+ -_p
-l
=m(H-EC)
(21)
+ (rp)@l with
m >. 0 aqd E, being the lowest eigenvalue of H. ObVio~~sIyMb is the sum of two positive semidefinite op.
eratjrs. Furthe~ore (flH.- E;olj’T,nnishes only for -f=:Q and according to section 2 (y I$‘,, > f 0. This implk; Al,-, > 0, where Mi’ is the wave operator given in eq. (112). Thus we have given an extiple for a wave : ,. ‘1’ ’ (I’@ -,op’e&or wk..wifi the property I+‘;’> @and in the.folto ,.:such A-1 l@-%?l(B-~ -A-‘$ ,,; ., Iow+g we restrict okelves .I wave operatqrs Wk’ : .’ .’ : .:-,-, ., .. “. :I,
‘CombiiGng(14) and (i5) we obtain the operator in-, ,A-? 5B8-1.
“.
bt us (15)(ii)
~(8-1-~-1)1~)(~16-i-A-lI~)-1(~~(B-l-~-~j.
equality
(i6).be,comes..
-(B-1A-I)IX)(XIAB-1A-AIX)-l(~I(AB-1-I), ‘. :,
(Ju,‘w,) 4. Lower bdunds
IqI)=AIX),
I being the identity [8] of
for ari arbitrary function .$.Equality holds for xOpt = A-l<. Identifying.4
1 ,March 975
..’ CHEMICAL: PHYSICS LETTERS
CHEMICAL PHYSlCS
Volume 31, number 2, For ‘vi1 > 0 a 4 > 0 exists with rv;l
>ql>O.
(22)
This implies
’
Wj+@~l
>o.
(23)
Hence we may choose ,4-l in (18) and obtain
B = qlVil
= I$,
lq-l rui3,- W,-2lX) (24)
The special choice of B enables one to use (5). With the aid
of (5) and (4) inequality
(24) becomes
I(rplrij;l-qlx)12 Isk12>q
1 -.
[ Considering ISol2 2 4
spite the difficulties in evaluating the integrals overH3, the cube of the hamiltonian operator [IO]. One easiIy sees that (26) and (27) require the same integrals, however in (26) E, does not occur. The lower bound (21) to lskl2 resembles Wang’s formula [9] , which holds for excited states
and .E= +D
(p,~~,9)~q-l~9,;uk,p) _‘(pf+il -r’xu2 (x
13farch 1975
LETTERS
where H, = H - Ek f . In (21) and (28) the same integrals occur but (28) contains the eigenudues Ekir, Ek, E k+ ]. whereas (21) conttins only Ek. However the quality of the bounds (21) and (26) depends upon the parameters m, p or 4 respectively. p and 4 are restricted by
-I
(x I IUi3- qrt~;21XJ 1
(25)
.
the special case ‘Vi1 = M,, (25) reads
[
1 - [I(cpIm(H-Eo)+IG44-
41x~121
x [(xl Em(H-E())+I9~(9113 -‘qjm(H-E())
+
lq~<$ol)lx~]-1
1 -1
-
(26)
5. Discussion The closed expression (4) for the overlap in terms of wave operators enabled us to derive upper and lower. bounds to the overlap by means of operator inequalities. The connection between ‘Lheinequalities (1 l), (20) and (25) is displayed by the fact that xopt = It$wwkq for each of them. It is remarkable that the lower bounds (21) and (26) contain only one eigenvalue of the harniltonian in contrast to other known variational lower bounds; see for instance the review of Weinhold [2] . Inequality (26) requires similar information as Wang’s lower bound [9]
It can be shown that for fmed m and x the bounds (21) and (26) are monotonous increasing functions of p and 4. The proof is elementary but rather lengthy and will be omitted here. The monotonicity of (21) and (26) implies that the better p,, or qO is approximated by p or q, the better will be the bounds to [Sk!2 for fied nz and x. Unfortunately the op&isation of the bounds with respect to m whith faed x is not straightfomard, since p or q must be varied simultaneously with nz. Moreover for every m the optimal p or Q has to be evaluated anew. The lower bounds p, q to pO, q,, can be determined for instance by the method of intermediate problems [ 111 or by Weinstein’s formula [ 121. The necessity to calculate the best available p or q for every ~tz separately suggests a trial and error procedure for the opttisation of the bounds (21) and (26) with respect to m. Certainly a more detailed study and discussion of the properties-of these bounds is necessary. Simplified nonvariational versions of the bounds derived in this paper.
’ OptimisItion
((9 l(H-- E())W
of (28) with respct
to the norm of x leads :o
E,)IX)12 ‘,
(27)
.’ which has been applied to the ground
state of He de219
.Vciume 3l;‘number
.‘_ . ‘.tid.the Tel&on toother bqimds are being investigated at present.
[2].,'
for the overlap
: W& wish to-thank Professor O.E:Polansky tid Dr. G. Ojbrich
..I:
for his for
:
References
: -.
[l] F. Weinhold;~Admn.
299.‘:
.-
‘. .. .
:. ‘,
.’ L
‘.
‘[12] D.H. Weinstein, Proc..I’&i Acad. Sci. US 20 (1934) 529.
-’
,,
1 March 19’,5
,.
1972).
Quantum Chem. 6 (1972)
,, ‘. .’
[2] F. Weird&d, J. Math. Phys. 11 (1970) 2127; [3] P.A:Braun and TX. Rebane, Intern. J. Quantum Chem. : 3 (1969) 1qsa. [4] F. Weinhold and P.S.C. Wang;-item. J. Quantum Chem. 5 (1971) 215: 1 [S ] P.-O..Lawdin, Rev.-Mod. Phys. 35 (1963) 702. and F. Mark, G-them. Phys. Letters 161 T: Hoffmann-Qstenhof 23 (1973) 302; -’ ; [7] P.-O. Lawdin, Intern..J. Quantum Chhem. 2’(1968) 867.. [S] P.-O. Lowdin, Intern. J. .Quantum (Them. 4 (1971) 231. [9] P.S.C. Wang, Intern. 5. Quantum Chem. 3 (1969) 57. [IO] F. Weinhold, J. Chem. Phys. 59 (1973) 355. .. [ 1 I] A. Weinstein and W. Stenger, hfethods of intermediate problems for eigenvalues (Academic Press, New York,
kcknowledgement ,, ..‘,
interest an.d encouragement stimulating discussions.
.:.
-,:
_ .
: ‘.
CHEMICAL PHYSICS IETT;ERs
2’
:
‘.
:
3 1, nukber
CHEMICAL
2
PHYSlC,S LETTERS
1 hkuch 1975
‘,
VARIATIONAL BOUNDS TO TIIE OVERLAP
”
Thomas HOFFMANN-OSTENHOF and Maria HOFFMA&%-OSTENHOF lnstihct fir Strchlencllemie Received 29 October
im Max-Flat@-Institur
fiir Kohlenforschung,
D-4330 hfiilheimjRuhr.
Germnjzy
1974
Starting from a closed expression
for the overlap, variational upper and lower bounds to the overlap ae derived by means
of operator inequalities.
1. Introduction
2.
The evaluation of upper and lower bounds to quantum mechanical properties [l] usually required the estimation of the overlap Sk =(~I~k);(~l~O)=(~kl~Clk) = 1; 50is an approximation to a solution ak of the lime independent Schrcdinger equation HGk = Ek $k (k = 0,1,2,. . .),ff being self-adjoint. niloreover lSk I itself is a measure of the accuracy achieved in an actual calculatioe [1,2]. The importance of variational bounds to the overlap stems from the fact that ISkI can be approximated in a systematic manner and equality can be,achieved at Least in prikiple [2]. These variational bounds may be written in the following way
For every function and lower
bounds
I(% x) 2 ISk 12,
x the function& respectiveIy
J(& x) =q
in (1) yield
upper
A closed expression for the overlap Let us define an operator Igk such that
‘@ = $kf$,
where we assume Sk + 0 and the corresponding eigenvalue Ek is non degenerate. Such an operator Ivk will be called wave operator [S, 51. From the deftition it follows immediately that IQ
= (,~llQu&)-l.
(2)
We can distinguish between two kinds of variational
bounds. (a) The equality sign holds only if x0 t = \clk [2-l] or.(b) x,, 5 Gk. In contrast to boun& of type (b), bounds of type (a) require a x which is a better ap-’ proximation tp Gk than rp, if high accuracy is desired. !n the foilowing we wti be concerned only with bounds : ,, oftype (b).
(4)
We note further that QlW&00) = 1.
(5)
Wave operators are often encountered in partitioning technique and perturbation theory [7]. In general these operators are not self-adjoint. However For the followirg derivations self-adjoint wzwz oyerators will be useful. Let M-k = rn(H-
12.
(3
m f 0
Ek) + ILp>(~I,
(6)
being a co&plax parameter. Obviously
MkGk = Q.
(7)
AssumingAIk non-singular and m real thenJi,-’ represents a self-adjoint wave operator. Substituting rVk= J%$’into (4) we obtain lSJ2
= c~lM~21~r’.
(8)
The problem is now to find b&n& to the expectation value in (4) or (8). Tkwill be done by means of oper‘ator inequalities.’
:
;. .‘,
277
Volume
3!, number’2
3. Upper bouqds
Substituting
to the overlap
..A-Jk3-l To d&iv& a vkiational upper bound stk from.the operator inequa!ity A-’ 2 1x1 c&I
to Isk I2 we. ,.
IXF~~l,
(9)
where_&> 0. x is ti arbitrary function and TX, (x]A Ix>-~’ (xl is called an inner projection A-l .‘Therefore
cf:;d :
i$? > 14 Ix$/(xlA
IX>
(10)
(10)
with ($Wk)-’ and {‘with 9, (4) and uppkr boynd
lead to the following
IS,12 ~~xl~rv,‘rv,~-‘Ix~/lclplX~12. Fo; the special wave operator
(10
Wk =ilI~‘, ‘.
w,=‘[m(H-Ek)+I~)(~(]-1
that is. td say ,’
.,
(9
minimisation of (11) with respect to the complex parameter m-produces ihe well known upper bound [9] (13)
to the overlap
The derivation oflower bounds to ISkI will again be based on an cperator inequaliiy. Let A and B represent self-adjoint operators with A > B > 0.Consider the identity
-/g-l =B”
L (B-1. - A-1).
(14)
If A > B > 0 it can be shown that B-l >A-’ > 0 [7] . Therefore 8-l - Am1 > 0.Hence we cti appl the Y method of inner pr,oj$ions [S] to A_1 - B- with an arbitraj function 77
-B-l,_/' ,, :.
”
‘. +-%4-‘)1;!~~18-1 . . ,‘, ~.?&c
I ‘,
-,.
,-,
”
..
operator.
-
(17)
That means
(~I~__l,,,.,,‘l,-l,,,_~lc~le-‘a~rlX,i~ (18) (y_IAB% -A lx> for ELIarbitrary function t, provided the corresponding integrals exist. Equality is achieved for x0 t =A $. Not& that xopt does not,depend upon B. 1Qe want-l to emphasize that for both variational principles (10) and’ (18) xopt turns out to be identical. Furthermore A-’ does not occur on the right hand side of (18). Inequality (18) will be the starting poini for the ‘derivation of lower bounds to ISk12. Identifying A with an appropriate chqice for the operator B (W,t WJl has to be made in order to apply (18). In the following we shall consider,two s&Al choices for B. i:) First we note that for .properly chosen non-singular IV-!l the operator (W+W )-I is positive definite and thkercfore a p > 0 e&s iuch that -1
>pI>O.
(19)
We put B = pl and E = cp and insert into (18). Taking (4) into account we obtain the variational lower bound to the overlap
I(#p,)-‘-plx)12
ISkI’ >p_ 1 [ For-the
cxl(rv,fru,)-*-p(Co,‘W,)-‘Ix
wave operator
IS&=P
[
1-
led_12 -PlX~121
I~o>cipl~4
_1
1
im(H-~~~f!~~~~l~~lX~l-~ ._ considerkb
. (20) J
(12) with real m this becomes
[lbpl &y)+
x~[cxI{m(H--q+ -_p
-l
=m(H-EC)
(21)
+ (rp)@l with
m >. 0 aqd E, being the lowest eigenvalue of H. ObVio~~sIyMb is the sum of two positive semidefinite op.
eratjrs. Furthe~ore (flH.- E;olj’T,nnishes only for -f=:Q and according to section 2 (y I$‘,, > f 0. This implk; Al,-, > 0, where Mi’ is the wave operator given in eq. (112). Thus we have given an extiple for a wave : ,. ‘1’ ’ (I’@ -,op’e&or wk..wifi the property I+‘;’> @and in the.folto ,.:such A-1 l@-%?l(B-~ -A-‘$ ,,; ., Iow+g we restrict okelves .I wave operatqrs Wk’ : .’ .’ : .:-,-, ., .. “. :I,
‘CombiiGng(14) and (i5) we obtain the operator in-, ,A-? 5B8-1.
“.
bt us (15)(ii)
~(8-1-~-1)1~)(~16-i-A-lI~)-1(~~(B-l-~-~j.
equality
(i6).be,comes..
-(B-1A-I)IX)(XIAB-1A-AIX)-l(~I(AB-1-I), ‘. :,
(Ju,‘w,) 4. Lower bdunds
IqI)=AIX),
I being the identity [8] of
for ari arbitrary function .$.Equality holds for xOpt = A-l<. Identifying.4
1 ,March 975
..’ CHEMICAL: PHYSICS LETTERS
CHEMICAL PHYSlCS
Volume 31, number 2, For ‘vi1 > 0 a 4 > 0 exists with rv;l
>ql>O.
(22)
This implies
’
Wj+@~l
>o.
(23)
Hence we may choose ,4-l in (18) and obtain
B = qlVil
= I$,
lq-l rui3,- W,-2lX) (24)
The special choice of B enables one to use (5). With the aid
of (5) and (4) inequality
(24) becomes
I(rplrij;l-qlx)12 Isk12>q
1 -.
[ Considering ISol2 2 4
spite the difficulties in evaluating the integrals overH3, the cube of the hamiltonian operator [IO]. One easiIy sees that (26) and (27) require the same integrals, however in (26) E, does not occur. The lower bound (21) to lskl2 resembles Wang’s formula [9] , which holds for excited states
and .E= +D
(p,~~,9)~q-l~9,;uk,p) _‘(pf+il -r’xu2 (x
13farch 1975
LETTERS
where H, = H - Ek f . In (21) and (28) the same integrals occur but (28) contains the eigenudues Ekir, Ek, E k+ ]. whereas (21) conttins only Ek. However the quality of the bounds (21) and (26) depends upon the parameters m, p or 4 respectively. p and 4 are restricted by
-I
(x I IUi3- qrt~;21XJ 1
(25)
.
the special case ‘Vi1 = M,, (25) reads
[
1 - [I(cpIm(H-Eo)+IG44-
41x~121
x [(xl Em(H-E())+I9~(9113 -‘qjm(H-E())
+
lq~<$ol)lx~]-1
1 -1
-
(26)
5. Discussion The closed expression (4) for the overlap in terms of wave operators enabled us to derive upper and lower. bounds to the overlap by means of operator inequalities. The connection between ‘Lheinequalities (1 l), (20) and (25) is displayed by the fact that xopt = It$wwkq for each of them. It is remarkable that the lower bounds (21) and (26) contain only one eigenvalue of the harniltonian in contrast to other known variational lower bounds; see for instance the review of Weinhold [2] . Inequality (26) requires similar information as Wang’s lower bound [9]
It can be shown that for fmed m and x the bounds (21) and (26) are monotonous increasing functions of p and 4. The proof is elementary but rather lengthy and will be omitted here. The monotonicity of (21) and (26) implies that the better p,, or qO is approximated by p or q, the better will be the bounds to [Sk!2 for fied nz and x. Unfortunately the op&isation of the bounds with respect to m whith faed x is not straightfomard, since p or q must be varied simultaneously with nz. Moreover for every m the optimal p or Q has to be evaluated anew. The lower bounds p, q to pO, q,, can be determined for instance by the method of intermediate problems [ 111 or by Weinstein’s formula [ 121. The necessity to calculate the best available p or q for every ~tz separately suggests a trial and error procedure for the opttisation of the bounds (21) and (26) with respect to m. Certainly a more detailed study and discussion of the properties-of these bounds is necessary. Simplified nonvariational versions of the bounds derived in this paper.
’ OptimisItion
((9 l(H-- E())W
of (28) with respct
to the norm of x leads :o
E,)IX)12 ‘,
(27)
.’ which has been applied to the ground
state of He de219
.Vciume 3l;‘number
.‘_ . ‘.tid.the Tel&on toother bqimds are being investigated at present.
[2].,'
for the overlap
: W& wish to-thank Professor O.E:Polansky tid Dr. G. Ojbrich
..I:
for his for
:
References
: -.
[l] F. Weinhold;~Admn.
299.‘:
.-
‘. .. .
:. ‘,
.’ L
‘.
‘[12] D.H. Weinstein, Proc..I’&i Acad. Sci. US 20 (1934) 529.
-’
,,
1 March 19’,5
,.
1972).
Quantum Chem. 6 (1972)
,, ‘. .’
[2] F. Weird&d, J. Math. Phys. 11 (1970) 2127; [3] P.A:Braun and TX. Rebane, Intern. J. Quantum Chem. : 3 (1969) 1qsa. [4] F. Weinhold and P.S.C. Wang;-item. J. Quantum Chem. 5 (1971) 215: 1 [S ] P.-O..Lawdin, Rev.-Mod. Phys. 35 (1963) 702. and F. Mark, G-them. Phys. Letters 161 T: Hoffmann-Qstenhof 23 (1973) 302; -’ ; [7] P.-O. Lawdin, Intern..J. Quantum Chhem. 2’(1968) 867.. [S] P.-O. Lowdin, Intern. J. .Quantum (Them. 4 (1971) 231. [9] P.S.C. Wang, Intern. 5. Quantum Chem. 3 (1969) 57. [IO] F. Weinhold, J. Chem. Phys. 59 (1973) 355. .. [ 1 I] A. Weinstein and W. Stenger, hfethods of intermediate problems for eigenvalues (Academic Press, New York,
kcknowledgement ,, ..‘,
interest an.d encouragement stimulating discussions.
.:.
-,:
_ .
: ‘.
CHEMICAL PHYSICS IETT;ERs
2’
:
‘.
: