- Email: [email protected]
Upper and lower bounds for expectation values
.Volume 11, number 3
:
CHEMICAL PHYSICS LETTERS
. 1.5 October 1971 :
.
.’
UPPER AND LOWER BOUNDS FOR’EXPECTATION VALUES P.&C. WANG ‘: Departme& of Chemistry, Simon Fraser Universi?y, Burnaby 2, B.C., OInada Received 30 July 1971
‘_New upper and lower boun& for quantum-mechanical
expectation values are d&ived from the positivity condiforms of those proposed recently by Mauiotti and Parr. for the inverses of the electron-nucleus and itItere&tlUIliC the helium atom and compared With &se of Jennings and VU-
tion for Gram determinants. These bounds are strengthened 4~ &iJl~~tr~tIve
cmnlpk
CITO~boun&
zuc cllcuhted
distance in thegraund and fist excited ‘S states of son, Weinhokl, and Mazziotti and Parr.
1. Introduction : h this letter we use the Gram determinant method of Wetiold [l] to derive a new error bound formula for the expecktion values in quantum-mechanical systems. These bounds improve the ones given recently by Mazziqtti and Parr [2] and-are applicable to essktially the same class of operators. Numeriwl applications to the ground and fust excited IS states of the He atom for the operators (I/r1 + I/rz) and l/r,,, where rl, r2 and r12 are the electron-nucleus and electron-electron distances, respectively, are made in section 3,and compared with the results obtained fi’om other known formulas.-
2. Deritiion
of the error bound formulas
For an arbitrary set of vectors (Yi), it is well known [3 3 that the Gram determinant is non-negative, D = det(olJv$ B 0 .
(1)
This inequality has been used successfully by Weir&old [4] t6 obtain errol bounds for various quantum-mechanical quantities. In the following, we shall use (1) to derive bounds for the expectation values of (hermitian) operators,F, for those F that are bounded (from below) relative to another (hermitian) operator C;, that is, there exist constants a, b tid c such that ** F’=_uFtbGtcXL
(2)
we shall u&e‘$ to denote exact (normalized) eigenfunction of the hamiltonian H, and x, 9 to denote approximate (normali?ed) digenfunctions. For the vectors @,F’x’and @,inequality (1) gives [I] *** (3) .‘_ .’ ,. ‘_ l Natioml’keaearch Coukil of Canada Postdoctoralfelbti.. .. i* This includes all positive and negativedefiite operators., l ** When ihe quantity in thegkrentheses ( )& negative it shodd be replmd.py Zero. .. -.
318 >‘. ‘I. ‘. .:. _’ ., ..‘, ‘,;. ,:’ .. ; ,. ‘. ‘..., :, .’ ,--. _ ‘_I .‘. .‘,. .... . ‘: ,. .. I ..- :_:
. ,. .:
: -I -, ..
., ,:’
.,, ‘_ ,.. ,. ._ -. .: ‘. .: _’ .. ,.;‘. ‘: ,,,.
:
.’.
-, -._
::.
..I
: _ ,.;,.... ,’ : 1, .. .; _.I ,,, .’ ., .;. ;, .‘_ :_: f :; ‘-,, ..,. .‘.,: .. ;, : ‘.’ “. :..__
.,‘,
Volume 1 I, number 3
CHEMICAL PHYSICS, LEVERS
ki October
1971
Byputting’GkH, x=#,a=p, h= 1 andc = -h @ is the ~owest’eigen~~ue of the operator H +@ or a Iower bound to it) in (3) we get the fohowing bounds to the expectation value of IF, (4) where the upper inequality holds for ~1k 0, the lower for CI> 0; E =
offf,
cB>= +z@llq% , A2(p,h;X,$)
=D(H;X,#J)
+p2D(F;X,q5)
+X2(1
-
i- ‘zI.I(Xfl~I(X-6))
- 2~~X@?t(X-@))
- 2;tL(X@?(X-#))
T
W;x,@ = {xJB21xJ - (xl%i2 Since the right-hand side of (4) decreases (if CL> 0) or increases (if cr < 0) for decreasing S’(O), we can use a fower bound S,(4) in pIace of S(G). SimiIarly we can replace the exact energy E by its upper bound (QJHJ$>in (4). Better but more compticated bounds can also be obtained from (3) by choosing x and Q to be diif‘erent,
where again, the upper inequalities are for the case ,u < 0, the lower ones for ct > 0 and EU(#) = fE --S&l “*. Since the second term on the right-hand side of (4}-(6) is aIways positive for @> 0 and rqptive for p < 0, these inequalities still hold when this term is !cft out, in which case (4)-(6) become the Mazziotti-Parr bounds [2]. Furthermore, for positive operatorsp, in the limit when ~1+ +QO,these new bounds reduce to those of Weinhold [ 1,5] . (This latter fact can also be seen more clearly from (3) by putting 1;1=1, b = c = 0,)
3. Applications In order to compare the bounds (4)--(6) with those given pre$ously by Jennings and Wilson [6j , We~hold [I ,5] and Mazziotti and Parr [2] , we appfied them to the operators*&’ = $(rcl +rzf) and FiJ for the ground and fast excited IS states of the helium atom. In this case, the lowest eigenvalue of N [email protected] be caicuIated fairly accurately from the (i/Z)-perturbation expansion for the ground state energy of twoebctron atoms, and these were used for X in &I computations** . For the ground state, we choose Q to be a product of hy&og&ic functions, 9 =Nexp(-&
-4Q
,
where **T f = 1.6875 and N is the normal.iz+ion constant. (All quantities are in &om.ic units.) For this #, the. EckaJ lower bound to the overlap [8] l(#JJl)J is 0.9623, the Weir&old lower bound is [9] 0.9870. In eqs. (5)‘and (6), two different functions are used for x, .
* The Symmetric form~&$ +ri”) is used here ,j.nstead of tither linear &mbi&tions since c&isoperator gives the smallest . . ~Iue forD(F;x,q$ ** The expression use~I in our campuht&-ms was the 1 Sterm expand& df S&err an&Knight f7], we only cntidered those g : .’ lying irrtficrange !JJ~< 2.0, therefore the perturbation expansion f&k is kid. *** ‘II&i is the value for f t&it ni,inirnizesthe energy expectation v&t? ~~I~)/~~, and errs &&rrrzeSt&eMstiotti-b .‘~~ds.~ ; ‘, ‘, .‘. ‘, : : ‘. ..,” . :.-. ‘,_: ‘. .” 319 .: .’ .,. .‘. _..’ .,.: ,_‘,’ .( ,. _. : . . . .. ;.’ ,: . ‘ /. ._. ._. .,’ ., ‘, ‘.,.‘,_ . . ,,. ‘1’ ., %’ ‘, .,,, .:. _ .’ ‘. ‘,“, ., ,. _.. .‘,’ ,’ ... :., ‘ , ..,, ” ..,, ‘, : . ..., ;_ ;.I.,. .,. -,, . _.,:. .:: : ,. .,., .L :. ‘:. I_ _. ,:i“ .. _:
..$.
.,
,I
.-
‘..
:..
;:
‘.
_,
;,,.-F
:
.,
‘.
ii\
l(ri’
.,
:
_..
,,: ,y.., ., ,I ., j .:. .., :,I,., I. :
.’
l.mb)
1.5428)
i*i::z
1.00
0.60
1.00 121
1.1523
1.1288“)
2.73
1.9251 c,d)
:’
112
,
d.568b)
.I -0.769@
::~:~
Jtknings‘Wilson IfA
0.616b)
1:nob)
i=+
1.1523
1.1288d)
2.1945a) 2.0632b)
1.92sld)
.,.
P_.._C,
0.8029 a) 0;6482 b)
1.1265")
1.1263")
0.8012 a) 0.6461 b)
1.4070 a)
x=x1
1.4065 a)
x=x1
WeinhoY [5]
a) SL($) = 0.9870 (Reinhold lower bound). b) SL(&j = 0.962?(Edtart lower bound). c) Corrected Mazziotti-Rur results. .’d) Best result for all p in the range 9 < p 6 2.0. e) E$art lower bound iswed for SL(~>_
L.;,:, ,, -’_, .,:..: ;,.. ,-.,. :; .,,I.:.’ ::’ ,, _,: :
'. ...
+;t)
0.24
1.1523
1.1288
2.1593a) z.ol34b)
1.9251
l.ES21
1.1288
2.1587a) 2,0131
1.9251,
I .0631
1.9042
1.8175
1.9098
__ 1.0335
1.1179
1.7651
1.8718
0.7373
0.7629d) 0:60
1.oo
0.68
(2)
1.4516Gd)
MazziottiParr [Zl
1.00
0.24
cc
0.7433 a) o.7373b)
0.7629d)
1.3935 a) 1.2522b).
1.4516d)
Formula (4)
0.7881 a) o.7373b)
0.7742 a) 0.7629$&
0.7649 a) 0.7629”) 0.7763 a) 0.7373b)
1.3337 a) 1.2025b)
1.4516
x=x;
1.3355 a) 1.2039
1.4516.
x=x1
Formch (5)
aj SL(&) = 0.9623 (E&art lower bound). b)SL(+) = 0.9870 (Weinhokl lower bound). c) .CorradedMazziotti-Parr results. Crhe values given in ref. [2] are in error.) d) Best result for all p in the range -2.0 < CI< 0. .. e) Eckart lower bound is used-foi SL(X). .: ,. Table 2 ‘. ... . Lower bounds for the expectation value of F in the ground state of the He atom
;
.,c:_ ., ~~.‘,..~(l;‘,+r~‘,, ','. ... : :,
,_ ”
,:,’
:
,
.’ .:
,, .., : : ,‘. .I .. :. I,.
.,
‘.
, .‘-
c .,
.,
:.’
~,,:
..‘,,
.’
:;...w ,, :; : ,. ; Table 1 .’ b&dsforjthe ,I. ;,, :,g :.:. .Uppc expectationvalues of F in the ground state of the He atom (exact expectation value [14] for i(r;’ +$) 7 .‘.‘. :,.,. ,,j. ,‘. .’ 1.6883,forr;: = 0.9458) ‘: ,: : .: : .. ?, .:,, ..‘., .,. ,:.I,:.;‘, ..’ Formula (5) Formula (6) e) ’Jennings+ Mazziotti‘I :.:: : ‘% ,‘, Formula (4) :..: ‘, : Wilson [6]. --’ pm I21 x=x1 x-x2 x=x1 'x=x2
‘,
.,
o*8o73
0.7824
.1-5482
1.4773
x=x1
0.8408
,0.8111.
1.5862
1.5038
-.
x=x2
Pormuh (6) @I
..
..
:
:
2 i ul .;i
:
G
E
I;; =3
B
B w z *
2
B
b-. Y
.$
Volumti 11, number 3
CHEMICAL PHYSICS LETTERS
15 October 1971
Table 3 Upper and lower bounds for the expectation value of I; in the Z’S state of the He atom Upper bounds
Lower bounds F
WFW
i(ri’ +rz’)
r;\
Weinhold
1.1343
0.2497
[II
Improved Weinbokl [S]
0.8219
0.8237
0.1616
MazziottiI21 b
(4)
-0.87
2.5597 a)
1.4606
0.8934
-0.98
2.5657
1.4603 a)
0.2011 a)
0.2011 a)
-2.00
1.6341 aI
0.3596
0.0337
0.1355
-1.53
1.6754
0.3587 a)
MazziottiPm I21
Formuh (4)
p
’ OS37
0.8168a)
0.8955 a)
1.00
0.8082
1.66 1.00
FormuIa
0.1709
a) Best value for IpI Q 2.0.
x1
=
[cl exp(-ar,
-br2) + c2r12 exp(-dr,
-dr2)]
+ symmetric
part ,
[IO]a = 1.3392, b - 2.3227, d = 1.7840. The energy expectation value for this function is -2.9023 14, the energy variance (x~KZ21x1>- (x1 Wlxl) 2 = 0.0509 17. The second function ~2 used is the six-term HylIeraas [ 1 l]
where
function with orbital exponent chosen to be 1.75676. This function has an energy expectation vaIue of -2.903329 and a variance 0.017389. The results obtained from (4)-(6) for these C$and x’s are shown in tables 1 and 2, where we also incIuded the corresponding Mazziotti-Parr, Jennings-Wilson, and Weinhold bounds for comparison. For the 2lS state, $ is chosen to be the nine-term function of James and Coolidge [12] with an energy value equal to -2.144047 and a variance 0.033727. The Weinberger formula [13] gives the value 0.98852 as a Iower bound
to the overlap
this state
of Q and the exact
are given in table
2’s
state waveftinction
of the He atomThe
results
obtained
From (4) for
3.
From these results we see that the new formulas give fairly tight bounds for this problem. The computations involved are quite simple since accurate expressions for A are available. In the general case, the difficulty in obtaining X is the major drawback of the present method as well as that proposed by Mazziotti and Parr [2]. Both these methods are of somewhat limited applicability since F has IO be relatively bounded with respect to the hamiltonian H, and thus cannot be used for many operaiors of physical interest *.
Acknowledgement The author is grateful to the referee for many helpful comments.
+ The author wishes to thank the referee for informing her that Mazziotti 1151 k~s recently extqded operators.
their formulas to wb0und-h
... ‘.
.; : .: ,.
.
321
: ,_: ..:
‘_
,‘, _;
:
;:
‘._
‘. -.
,. ‘1 .. -,.. ,’ ,:. .P’.. .‘., : : 8 -: : ,,’ ....‘. ,- .,,_-.,: ./ _., -;. ,. .I.
.,
..’
_
,.jGlme11,&nber ._.:
._
CHEMICALPHYSICS
3
LETTERS
..,
Ref~ences..
‘:.
[I] I&SIIIKM,
J:Phys. Al (1968) 305: [2] A.‘Mazzidttiand R.G. Parr, j. Chcm. Phys. 52 (1970) 1605. [3] L. Mirsky, An intrtiduction to linear a$ebra (Otiord Univ. Press,.Eondon, 1955): [4] F. Wei+@d, Phys. Rev. Letters 25 (1970) 907 and references therein. [S] F., Weinh?ld, J. Phys. Al (1968) 535; Phys. Rev. Al (1970) 122. [6] P:Jennings and E.B. Wilson, J. Chem. Phys. 45 (1966) 1847; 47 (1967) 2130. [7] ,Scherr ana Knight;Rev. Mod. Phys. 35 (i963) 436. .[8] C. Eckart, whys. Rev. 36 (1930) 878. [9] F.,We,jnhokl, J. Chem. Phys. 46 (1967) 2448. 1[lCl] R;A’. Bonham and D.A. Kohl; J. Chem. Phys. 45 (1966) 2471. .,[11] AD. Budcingham and P.G.‘Hubbard, Symp. Faraday Sot. 2 (1968) 41. 1121 &S. Coolidge and H.M. James, Phys. Rev. 49 (1936) 676. [13]-H.F. Weinberger; J. Res. Natt Bur. Std. (U.S.) 64B (1960) 217. [14] CL. Pekaris,Phys Rev. 115 (1959) 1216. [15] A. Mazziotti, J. qem. Phys., to be published.
--
:
J5 October 1971
:
CHEMICAL PHYSICS LETTERS
. 1.5 October 1971 :
.
.’
UPPER AND LOWER BOUNDS FOR’EXPECTATION VALUES P.&C. WANG ‘: Departme& of Chemistry, Simon Fraser Universi?y, Burnaby 2, B.C., OInada Received 30 July 1971
‘_New upper and lower boun& for quantum-mechanical
expectation values are d&ived from the positivity condiforms of those proposed recently by Mauiotti and Parr. for the inverses of the electron-nucleus and itItere&tlUIliC the helium atom and compared With &se of Jennings and VU-
tion for Gram determinants. These bounds are strengthened 4~ &iJl~~tr~tIve
cmnlpk
CITO~boun&
zuc cllcuhted
distance in thegraund and fist excited ‘S states of son, Weinhokl, and Mazziotti and Parr.
1. Introduction : h this letter we use the Gram determinant method of Wetiold [l] to derive a new error bound formula for the expecktion values in quantum-mechanical systems. These bounds improve the ones given recently by Mazziqtti and Parr [2] and-are applicable to essktially the same class of operators. Numeriwl applications to the ground and fust excited IS states of the He atom for the operators (I/r1 + I/rz) and l/r,,, where rl, r2 and r12 are the electron-nucleus and electron-electron distances, respectively, are made in section 3,and compared with the results obtained fi’om other known formulas.-
2. Deritiion
of the error bound formulas
For an arbitrary set of vectors (Yi), it is well known [3 3 that the Gram determinant is non-negative, D = det(olJv$ B 0 .
(1)
This inequality has been used successfully by Weir&old [4] t6 obtain errol bounds for various quantum-mechanical quantities. In the following, we shall use (1) to derive bounds for the expectation values of (hermitian) operators,F, for those F that are bounded (from below) relative to another (hermitian) operator C;, that is, there exist constants a, b tid c such that ** F’=_uFtbGtcXL
(2)
we shall u&e‘$ to denote exact (normalized) eigenfunction of the hamiltonian H, and x, 9 to denote approximate (normali?ed) digenfunctions. For the vectors @,F’x’and @,inequality (1) gives [I] *** (3) .‘_ .’ ,. ‘_ l Natioml’keaearch Coukil of Canada Postdoctoralfelbti.. .. i* This includes all positive and negativedefiite operators., l ** When ihe quantity in thegkrentheses ( )& negative it shodd be replmd.py Zero. .. -.
318 >‘. ‘I. ‘. .:. _’ ., ..‘, ‘,;. ,:’ .. ; ,. ‘. ‘..., :, .’ ,--. _ ‘_I .‘. .‘,. .... . ‘: ,. .. I ..- :_:
. ,. .:
: -I -, ..
., ,:’
.,, ‘_ ,.. ,. ._ -. .: ‘. .: _’ .. ,.;‘. ‘: ,,,.
:
.’.
-, -._
::.
..I
: _ ,.;,.... ,’ : 1, .. .; _.I ,,, .’ ., .;. ;, .‘_ :_: f :; ‘-,, ..,. .‘.,: .. ;, : ‘.’ “. :..__
.,‘,
Volume 1 I, number 3
CHEMICAL PHYSICS, LEVERS
ki October
1971
Byputting’GkH, x=#,a=p, h= 1 andc = -h @ is the ~owest’eigen~~ue of the operator H +@ or a Iower bound to it) in (3) we get the fohowing bounds to the expectation value of IF, (4) where the upper inequality holds for ~1k 0, the lower for CI> 0; E =
offf,
cB>= +z@llq% , A2(p,h;X,$)
=D(H;X,#J)
+p2D(F;X,q5)
+X2(1
-
i- ‘zI.I(Xfl~I(X-6))
- 2~~X@?t(X-@))
- 2;tL(X@?(X-#))
T
W;x,@ = {xJB21xJ - (xl%i2 Since the right-hand side of (4) decreases (if CL> 0) or increases (if cr < 0) for decreasing S’(O), we can use a fower bound S,(4) in pIace of S(G). SimiIarly we can replace the exact energy E by its upper bound (QJHJ$>in (4). Better but more compticated bounds can also be obtained from (3) by choosing x and Q to be diif‘erent,
where again, the upper inequalities are for the case ,u < 0, the lower ones for ct > 0 and EU(#) = fE --S&l “*. Since the second term on the right-hand side of (4}-(6) is aIways positive for @> 0 and rqptive for p < 0, these inequalities still hold when this term is !cft out, in which case (4)-(6) become the Mazziotti-Parr bounds [2]. Furthermore, for positive operatorsp, in the limit when ~1+ +QO,these new bounds reduce to those of Weinhold [ 1,5] . (This latter fact can also be seen more clearly from (3) by putting 1;1=1, b = c = 0,)
3. Applications In order to compare the bounds (4)--(6) with those given pre$ously by Jennings and Wilson [6j , We~hold [I ,5] and Mazziotti and Parr [2] , we appfied them to the operators*&’ = $(rcl +rzf) and FiJ for the ground and fast excited IS states of the helium atom. In this case, the lowest eigenvalue of N [email protected] be caicuIated fairly accurately from the (i/Z)-perturbation expansion for the ground state energy of twoebctron atoms, and these were used for X in &I computations** . For the ground state, we choose Q to be a product of hy&og&ic functions, 9 =Nexp(-&
-4Q
,
where **T f = 1.6875 and N is the normal.iz+ion constant. (All quantities are in &om.ic units.) For this #, the. EckaJ lower bound to the overlap [8] l(#JJl)J is 0.9623, the Weir&old lower bound is [9] 0.9870. In eqs. (5)‘and (6), two different functions are used for x, .
* The Symmetric form~&$ +ri”) is used here ,j.nstead of tither linear &mbi&tions since c&isoperator gives the smallest . . ~Iue forD(F;x,q$ ** The expression use~I in our campuht&-ms was the 1 Sterm expand& df S&err an&Knight f7], we only cntidered those g : .’ lying irrtficrange !JJ~< 2.0, therefore the perturbation expansion f&k is kid. *** ‘II&i is the value for f t&it ni,inirnizesthe energy expectation v&t? ~~I~)/~~, and errs &&rrrzeSt&eMstiotti-b .‘~~ds.~ ; ‘, ‘, .‘. ‘, : : ‘. ..,” . :.-. ‘,_: ‘. .” 319 .: .’ .,. .‘. _..’ .,.: ,_‘,’ .( ,. _. : . . . .. ;.’ ,: . ‘ /. ._. ._. .,’ ., ‘, ‘.,.‘,_ . . ,,. ‘1’ ., %’ ‘, .,,, .:. _ .’ ‘. ‘,“, ., ,. _.. .‘,’ ,’ ... :., ‘ , ..,, ” ..,, ‘, : . ..., ;_ ;.I.,. .,. -,, . _.,:. .:: : ,. .,., .L :. ‘:. I_ _. ,:i“ .. _:
..$.
.,
,I
.-
‘..
:..
;:
‘.
_,
;,,.-F
:
.,
‘.
ii\
l(ri’
.,
:
_..
,,: ,y.., ., ,I ., j .:. .., :,I,., I. :
.’
l.mb)
1.5428)
i*i::z
1.00
0.60
1.00 121
1.1523
1.1288“)
2.73
1.9251 c,d)
:’
112
,
d.568b)
.I -0.769@
::~:~
Jtknings‘Wilson IfA
0.616b)
1:nob)
i=+
1.1523
1.1288d)
2.1945a) 2.0632b)
1.92sld)
.,.
P_.._C,
0.8029 a) 0;6482 b)
1.1265")
1.1263")
0.8012 a) 0.6461 b)
1.4070 a)
x=x1
1.4065 a)
x=x1
WeinhoY [5]
a) SL($) = 0.9870 (Reinhold lower bound). b) SL(&j = 0.962?(Edtart lower bound). c) Corrected Mazziotti-Rur results. .’d) Best result for all p in the range 9 < p 6 2.0. e) E$art lower bound iswed for SL(~>_
L.;,:, ,, -’_, .,:..: ;,.. ,-.,. :; .,,I.:.’ ::’ ,, _,: :
'. ...
+;t)
0.24
1.1523
1.1288
2.1593a) z.ol34b)
1.9251
l.ES21
1.1288
2.1587a) 2,0131
1.9251,
I .0631
1.9042
1.8175
1.9098
__ 1.0335
1.1179
1.7651
1.8718
0.7373
0.7629d) 0:60
1.oo
0.68
(2)
1.4516Gd)
MazziottiParr [Zl
1.00
0.24
cc
0.7433 a) o.7373b)
0.7629d)
1.3935 a) 1.2522b).
1.4516d)
Formula (4)
0.7881 a) o.7373b)
0.7742 a) 0.7629$&
0.7649 a) 0.7629”) 0.7763 a) 0.7373b)
1.3337 a) 1.2025b)
1.4516
x=x;
1.3355 a) 1.2039
1.4516.
x=x1
Formch (5)
aj SL(&) = 0.9623 (E&art lower bound). b)SL(+) = 0.9870 (Weinhokl lower bound). c) .CorradedMazziotti-Parr results. Crhe values given in ref. [2] are in error.) d) Best result for all p in the range -2.0 < CI< 0. .. e) Eckart lower bound is used-foi SL(X). .: ,. Table 2 ‘. ... . Lower bounds for the expectation value of F in the ground state of the He atom
;
.,c:_ ., ~~.‘,..~(l;‘,+r~‘,, ','. ... : :,
,_ ”
,:,’
:
,
.’ .:
,, .., : : ,‘. .I .. :. I,.
.,
‘.
, .‘-
c .,
.,
:.’
~,,:
..‘,,
.’
:;...w ,, :; : ,. ; Table 1 .’ b&dsforjthe ,I. ;,, :,g :.:. .Uppc expectationvalues of F in the ground state of the He atom (exact expectation value [14] for i(r;’ +$) 7 .‘.‘. :,.,. ,,j. ,‘. .’ 1.6883,forr;: = 0.9458) ‘: ,: : .: : .. ?, .:,, ..‘., .,. ,:.I,:.;‘, ..’ Formula (5) Formula (6) e) ’Jennings+ Mazziotti‘I :.:: : ‘% ,‘, Formula (4) :..: ‘, : Wilson [6]. --’ pm I21 x=x1 x-x2 x=x1 'x=x2
‘,
.,
o*8o73
0.7824
.1-5482
1.4773
x=x1
0.8408
,0.8111.
1.5862
1.5038
-.
x=x2
Pormuh (6) @I
..
..
:
:
2 i ul .;i
:
G
E
I;; =3
B
B w z *
2
B
b-. Y
.$
Volumti 11, number 3
CHEMICAL PHYSICS LETTERS
15 October 1971
Table 3 Upper and lower bounds for the expectation value of I; in the Z’S state of the He atom Upper bounds
Lower bounds F
WFW
i(ri’ +rz’)
r;\
Weinhold
1.1343
0.2497
[II
Improved Weinbokl [S]
0.8219
0.8237
0.1616
MazziottiI21 b
(4)
-0.87
2.5597 a)
1.4606
0.8934
-0.98
2.5657
1.4603 a)
0.2011 a)
0.2011 a)
-2.00
1.6341 aI
0.3596
0.0337
0.1355
-1.53
1.6754
0.3587 a)
MazziottiPm I21
Formuh (4)
p
’ OS37
0.8168a)
0.8955 a)
1.00
0.8082
1.66 1.00
FormuIa
0.1709
a) Best value for IpI Q 2.0.
x1
=
[cl exp(-ar,
-br2) + c2r12 exp(-dr,
-dr2)]
+ symmetric
part ,
[IO]a = 1.3392, b - 2.3227, d = 1.7840. The energy expectation value for this function is -2.9023 14, the energy variance (x~KZ21x1>- (x1 Wlxl) 2 = 0.0509 17. The second function ~2 used is the six-term HylIeraas [ 1 l]
where
function with orbital exponent chosen to be 1.75676. This function has an energy expectation vaIue of -2.903329 and a variance 0.017389. The results obtained from (4)-(6) for these C$and x’s are shown in tables 1 and 2, where we also incIuded the corresponding Mazziotti-Parr, Jennings-Wilson, and Weinhold bounds for comparison. For the 2lS state, $ is chosen to be the nine-term function of James and Coolidge [12] with an energy value equal to -2.144047 and a variance 0.033727. The Weinberger formula [13] gives the value 0.98852 as a Iower bound
to the overlap
this state
of Q and the exact
are given in table
2’s
state waveftinction
of the He atomThe
results
obtained
From (4) for
3.
From these results we see that the new formulas give fairly tight bounds for this problem. The computations involved are quite simple since accurate expressions for A are available. In the general case, the difficulty in obtaining X is the major drawback of the present method as well as that proposed by Mazziotti and Parr [2]. Both these methods are of somewhat limited applicability since F has IO be relatively bounded with respect to the hamiltonian H, and thus cannot be used for many operaiors of physical interest *.
Acknowledgement The author is grateful to the referee for many helpful comments.
+ The author wishes to thank the referee for informing her that Mazziotti 1151 k~s recently extqded operators.
their formulas to wb0und-h
... ‘.
.; : .: ,.
.
321
: ,_: ..:
‘_
,‘, _;
:
;:
‘._
‘. -.
,. ‘1 .. -,.. ,’ ,:. .P’.. .‘., : : 8 -: : ,,’ ....‘. ,- .,,_-.,: ./ _., -;. ,. .I.
.,
..’
_
,.jGlme11,&nber ._.:
._
CHEMICALPHYSICS
3
LETTERS
..,
Ref~ences..
‘:.
[I] I&SIIIKM,
J:Phys. Al (1968) 305: [2] A.‘Mazzidttiand R.G. Parr, j. Chcm. Phys. 52 (1970) 1605. [3] L. Mirsky, An intrtiduction to linear a$ebra (Otiord Univ. Press,.Eondon, 1955): [4] F. Wei+@d, Phys. Rev. Letters 25 (1970) 907 and references therein. [S] F., Weinh?ld, J. Phys. Al (1968) 535; Phys. Rev. Al (1970) 122. [6] P:Jennings and E.B. Wilson, J. Chem. Phys. 45 (1966) 1847; 47 (1967) 2130. [7] ,Scherr ana Knight;Rev. Mod. Phys. 35 (i963) 436. .[8] C. Eckart, whys. Rev. 36 (1930) 878. [9] F.,We,jnhokl, J. Chem. Phys. 46 (1967) 2448. 1[lCl] R;A’. Bonham and D.A. Kohl; J. Chem. Phys. 45 (1966) 2471. .,[11] AD. Budcingham and P.G.‘Hubbard, Symp. Faraday Sot. 2 (1968) 41. 1121 &S. Coolidge and H.M. James, Phys. Rev. 49 (1936) 676. [13]-H.F. Weinberger; J. Res. Natt Bur. Std. (U.S.) 64B (1960) 217. [14] CL. Pekaris,Phys Rev. 115 (1959) 1216. [15] A. Mazziotti, J. qem. Phys., to be published.
--
:
J5 October 1971