- Email: [email protected]
Solution of the hartree-fock first-order equation for the helium atom
Volume 6. number 3
CHEMXAL
SOLUTION
OF
THE
LETTERS
HARTREE-FOCK FOR -
Battelle
PHYSICS
THE
HELIUM
1
FIRST-ORDER
August1970
EQUATION
ATOM
NICHOLAS W. WINTER Institute, Columbus. Ohio 43201, USA
kemonizl
and VINCENT McKOY and ARTHUR LAFERRIERE Arthur Amos Noyes Laboratory of Chemical Physics. California Institute of Technology, Pasadena. California 91109.
USA
Received 26 May 1970 The first-order equation for the ground state of the helium atom has been solved numerically. The perturbation energies and severat one-electron properties are Dresented and compared to variational results. Since the method easily with this approach.
is not variational,
the first-order
soiutions
for excited-states
can be obtained
1. INTRODUCTION SinanoElu [l] has demonstrated that the first-order wavefunction for the many-electron system can be expressed in terms of pair functions, each of which satisfies a differential equation similar to the first-order equations for the states of the two-electron atom. Previous attempts to solve this type of equation have been with variational methods [2]. In these cases one-is faced with the choice of a trial function and the evaluation of difficult integrals over interelectronic coordinates. It is possible to avoid these by solving the differential equation directly. This can be accomplished with high accuracy using the matrix finite-difference (MFD) method. The MFD method has been successfully applied to the hydrogenic pair equationsfor the I$, ls2s, and ls2p configurations [3], but it has not been previously demonstrated for non-local zero-order hamiltonians. Using the helium atom as a model for the more general pair equation, the first-order wavefunction was found for the Hartree and the Hartree-Fock hamiltonians. In the first section the perturbation equations are reduced to their partial-wave components, and the expressions for the perturbation energies are derived. The mmerical procedure is described briefly in the next section, and the results are then compared to previous variational calculations. 2. PARTIAL WAVE REDUCTION OF THE FIRST-ORDER EQUATION The zero-order ‘to = i(I)@(2)
wavefunction for the ground state is (ols - &4/n,
where the orbital @ satisfi& hH_@= [-iv2 and H&tree-Fock
-Z/?l
(1) both the Hartree
+J(l)]@
= $+’
(2)
~ -
: hHF~.=[~BVZ-Z/r;+~(l)-~(2)]~ equations with the usual definitions.
YE;m
(3) .(
V&me
6, nuder
3
CHEMICAL PHYSICS LETTERS
.: --.
41)
&l);(2)
*)d~~;
= ( j-+(2) $-f(2)
The first-order [Ml)
= J;Pw
dr,)$(l)
1
El -+7(W(2)
- 2s] @l(rlY2012).=
.-q
.
equations are
+M2)
1 August 1970
-.
(4)
@(W(2)
and
1 [hHF(l) +hHF(2)- 2~1*10’1@1,] = LE1
.
where
(5)
1
-&+2J(1)-K(:)+2c7(2)-K(2,
$(l)@(2),
El = j-@2(1)L@2(2)drldr2 ‘12 is the first-order energy. In general, the functions +I and P1 depend cn several variables and eqs. (4) and (5) cannot & solved conveniently in this form. By expanding the wavefunctions as follows, q(y1y2Y2) %cosQ1i),
*l&2812)
= T - yly2
(6)
it is possible to reduce the first-order equation to an infinite set of uncoupled differential the functional coefficients q(rlY2). The 2th member of this set has the form
equations for
[-;(~+~)-~-=*~+~+~Yl,+~(;~)_2~]~z~ylY2)
2
1
2 2 +-< QJ‘- Yl+ 1 I ~&-lPlJ9-2)
= [ L91+&9+J(y2)] for the. Hartree case and [
22 _- 1- _+_ 2( JYf
a2 aY;
>
_.z_z ?1
t w Y2
1
f 249-l) + 2J(?-2) - 2E
+ E(Z+ 2y;
.2y;
(7)
>
lq(YlY2)
YE =[[Jq+J(q)+J(Q)] 610 --& q&-ljP&-$ +F&Y~~~)P~&I) f q(y2y1)~ls(~21
I
for the Hartree-Fock
case, where
&!(7-19.2) = -L 2z+ 1 .j-P1&3)
P-3)
y>
-$&1~3)
dy3
and r,:= max(r2r3) ;
r< =. min (Y2r3) .
:. Once the functional-coefficients -pres&on
are.fouud,
the second-order
energy can be evaIuated from the ex.-
_-
x
:-
._ with
an
identical expression
for: the Hartree
-
;
case. The. third-order.
“: qj-,=. &!I /IQ I&> -~‘.E&f~ j*‘l>.L .i&&l), __-.. _-. _ . .. . ‘. . _. .. _: ‘_ -: :,,i_:.-: ‘1’76 : __ .,, .: ,I’_ : _.<.- ._-: _:-: _,: , ,. . ._. -_ _.... .. ..;‘. .:. 1 _
-q@(q)
Y -,
_-
_.
+4y2)])
Qqy2) :
energy -is found’f&
‘- [
:
do dy2
,
(9)
._
”
(ida) : -. .: ., .-;. -._.. y:_._ .I..: _i_yy _-:,:_.:--:;._ : ;_, ~_~,,__,‘.~__..‘~___ I-‘._.: __. .‘.. -:-.__-. _- :_:. . . . -..._ ,_ ,-_...-_ ,‘L ..-. ; .:I ._ _..‘. .,.__-, .. ‘;.. . . ..^ ._ _,
I
..
._ x
Volume 6. number 3
CHEMICALPHYSICSLETTERS
which Ieads to the following
f August 1970
two forms, k
= Z F k ckU, Z’) &rlr2) ? ’
+-tree)
-&(y1~z)d~lcb-2 y>
(lob)
2 Eg(Ha.rtree-Fock)
= I F k ck(Z, I’) lul(rLr2) , 9
*
WlY2)
drl d72
?->
ww
3. NUMERICAL ANALYSIS As noted previously, the solutions to eqs. (4) and (5) were found with the MFD method. This involved replacing the derivatives by a difference approximation and solving the resulting set of linear equations. The calcultitions reported here were carried out with the fourth difference approximation,
The difference truncation error [the neglect of higher differences in deriving (II)]was removed by Richardson [4] extrapolation. In order that the method could be applied to the more diffuse pairs such as excited states and negative ions, the coordinates were transformed as follows: y1
=x;;
Y =x2 2 2’
On a grid in x1 and x2, the cutoff at which the functional coefficient is required to vanish can be taken sufficiently large without adversely affecting the grid spacing. The results presented in the next section were extrapolated from grids containing 20, 25, 30, and 35 strips with a cutoff at rL or 1~ = 9 au. 4. RESULTS AND DISCUSSION The zero-order solution was taken as Clementi’s [5] five-term expansion. This_ gives A?&= -1.83592 au and E 1 = -1.02576au; .The pareal wave contributions to the second-order energy for the Kartree and HsrtreeiFock calculations are compared’to the variational results of Byron and Joachain [S] in tabIe 1. The MFD results for the I _=0 wave are estimated to be above the -exact value by about 0.000002 au.. The higher partial waves are-more abcurate. .Part of the discrepancy bekeen the variational and numerical results can be.attributed to the use of different zero-order functions, Overall the agreement is quite ‘reasonable. In table 2, the total perturbation energies arecompared to those of Byron and Joachain [6,7] tid of Weiss and-Martin [2]; The calcujations in the thtid-and fourth-colu&s were carried out ‘-1with Hylleiaas trial functions -rather .tha.n by a pa.rtiali.wave ezpansion..~The disagreement between the -, ,. . . ._._ _I, _._:..., _,: -.: .- . -177 . . . .: ..’ ., -_ ._. -.
Volume 6. number 3
CHZMICAL
Comparison
PRYSICS LETTERS
of the partial-wave
1’;
ccdributions
to E2.
Hartree-Fock b
-E2 (0)
-
MFD
Rgrtree
Variztiond
-0.013498
1 August 1970 ;
2
a)
MFD
-0.01347
-0.018022-
:
Variational a) -0.01798
E2(1)
-0.018980
-0.01894
-0.024800
-0.02475
E2 (2)
-0.003193
-0.00317
-0.003676
-0.00365
E2(3)
-0.000932
-0.00092
-0.001014
-0.00100
E2(4) E2 (5)
-0.000360
-0.00035
-0.000381
-0.00038
-0.000116
-0.00016
-0.000172
-0.00017
E2 (6)
-O.Z)OOwG
-0.00008
-0.000088
-0.00009
E2h
-0.O30048
-0.000049
E2 N!
-0.000029
-0.000029
E2 (9,
-0.OOOO3.3
E2
-0.037305
, ‘_ i
~0.000019
I -0.03725
-0.048250
-0.04817
a) Ref. 161: Table 2 Comparison of the total perturbation energies MFD
Ref. [S]
Byron Ref. [7]
Weiss Ref. [2]
-0.04604
-0.04835
Hartr.?e E2 E3
-0.048250 OIOO731~
-0.04817 0.0071?
E
-2.902617
-2.90267
=2
-0.037311
-0.03725
E3 E
-0.003685 -2;902656
0.00713
0.00741
-2.90258
-2.90262
-0.03719
___
-0.00377
-0.00346
---
-2.90269
-2.90232
w-m
Hartree-Z’ock
MFD results and those of Weiss is due mainly to the neglect of higher partial waves. Byron and Joachain [S] have estimated the contribution to E2 from partial waves with I 3 ‘7 to be -0.00915 au. This would in,dicate at-least an error of -0.00006 au in the MFD value due ta truncation at Z = 9. The corrected value is -0.04331~au which compares- better with that of Weiss. The third-order energy also suffers-from the truncation. The total energies do not compare unfavorably wi:h the exact value [8] of -2.9037 au or the best CI value [9] of -2.9032 au. In order to characterize the first-order solution, several one-electron properties were computed and are presented in table 3. In parallel with the contributions .to E2, the Hartree-Fock solution has considerably smaller expectation values than the Hartree solution. The agreement is better for the higher partial waves. For the exact first-order wavefunction of a closed.-shell state, the.firsGorder’corrections to the expectation values, (@O~*@I>, are zero. The amount tr, whjch they do not.vanish is a measure of the accuracy of the wavefunction and the extrapolation procedure. The quantities (j-) and (y_2>are both. improved trj’the second-order correction; however, the expectation value of l/r is bronght into spme,what worse agreement.with the exact -mlue. .., The re@ts presented here have ahown.that the Hartree. a.ud Hartree-Fock pair equations can be SC&& accurately aud efficiently with the MFD method: Bi+use it&.not va+tio&il, this approach w&i be most useful when applied to excited states._ Calculations,are in progress on the its2 IS autoionizing states of 3he two-electron atom usiing zero-order solutions sjmilar to those iecently.published [lO];..The the ground_state apply without a:bdjficatiop to thesy configui&ong. ‘. _. ,.-( .. ., ._. .:,_ _:-f.. ::_ :. ,,:-: ,I,’ ‘. ..:_ _:‘‘I ; .’ ..^. I .. ,‘-.
Volume
6. number 3
CHEMICAL
Partial-wave
PHYSICS
analysis
LETTERS
Table 3 of one-electron
f August 1970
properties Hartree-Fock
Hartree P (l/r)
t
w
(l/r)
+->
~~
0
0.017601
0.013216
9.927072
0.009356
0.006283
0.012363
1
0.012350
0.012460
0.017951
0.006890
0.006066
0.007781
2
0.000747
o.oou875
0.001228
0.000539
0.000575
0.000772
3
0.000113
0.000141
0.000201
0.000092
0.000108
0.000149
4
0.000027
Q.OOOQ35
0.000050
0.000023
0.000029
o.oooo4c
5
0.000008
9.000011
0.000016
0.000006
0.000010
0.0000t4
6
0.000003
0.000004
0.000006
0.000003
0.000004
0.000005
0.030851
0.026745
0.045627
0.016923
0.013077
0.021129
.@% C%lFl*k3
-0.00006
-0.00001
-0.00001
-0.00001
-0.aaa04
<*fJ IfiI*fj
3.3748
1.8546
2.3696
3.3748
1.8546
2.3696
TOW
3.3693
1.8613
2.3895
3.3727
1.8572
2.3773
Exact a)
3.3766
1.8589
2.3970
3.3766
I.8589
2.3870
(*I
0.010776
1%)
a) ref.
0.00001
0.005637
[S].
REFERENCES 0. Sinanoj$lu, Advan. Chem. Phys. 14 (1968) 237: Proc. Roy. Sot. A260 (1961) 379. A. W. Weiss and J. B. Martin, Phys. Rev. 132 (l963) 2116. N.W. Winter, Ph. D. Thesis, California Institute of Technology (1970). L. Richardson and J. Gaunt, Trans. Roy. Sot. A226 (1927) 299. E. Clementi, IBM 5. Res. Develop. Suppl. 9 (1965) 2. F. W. Byron and C.J. Joachain. Phys. Rev. 157 (1967) 1. F. W-Byron and C. J. Joachain, Phys. Rev. 146 0966) 1. C. L. Pe’keris, Phys. Rev. 126 (1962) 1470. A. W.Weiss, Phys.Rev. 122 (1961) 1826. D. E. Ramaker, R. B. Stevenson and D.M. Schrader, J. Chem. Phys. 51 (1969) 5276.
. _-
_.
._ .
..
.
: _,_
179
.:’ .,_ _.
-
CHEMXAL
SOLUTION
OF
THE
LETTERS
HARTREE-FOCK FOR -
Battelle
PHYSICS
THE
HELIUM
1
FIRST-ORDER
August1970
EQUATION
ATOM
NICHOLAS W. WINTER Institute, Columbus. Ohio 43201, USA
kemonizl
and VINCENT McKOY and ARTHUR LAFERRIERE Arthur Amos Noyes Laboratory of Chemical Physics. California Institute of Technology, Pasadena. California 91109.
USA
Received 26 May 1970 The first-order equation for the ground state of the helium atom has been solved numerically. The perturbation energies and severat one-electron properties are Dresented and compared to variational results. Since the method easily with this approach.
is not variational,
the first-order
soiutions
for excited-states
can be obtained
1. INTRODUCTION SinanoElu [l] has demonstrated that the first-order wavefunction for the many-electron system can be expressed in terms of pair functions, each of which satisfies a differential equation similar to the first-order equations for the states of the two-electron atom. Previous attempts to solve this type of equation have been with variational methods [2]. In these cases one-is faced with the choice of a trial function and the evaluation of difficult integrals over interelectronic coordinates. It is possible to avoid these by solving the differential equation directly. This can be accomplished with high accuracy using the matrix finite-difference (MFD) method. The MFD method has been successfully applied to the hydrogenic pair equationsfor the I$, ls2s, and ls2p configurations [3], but it has not been previously demonstrated for non-local zero-order hamiltonians. Using the helium atom as a model for the more general pair equation, the first-order wavefunction was found for the Hartree and the Hartree-Fock hamiltonians. In the first section the perturbation equations are reduced to their partial-wave components, and the expressions for the perturbation energies are derived. The mmerical procedure is described briefly in the next section, and the results are then compared to previous variational calculations. 2. PARTIAL WAVE REDUCTION OF THE FIRST-ORDER EQUATION The zero-order ‘to = i(I)@(2)
wavefunction for the ground state is (ols - &4/n,
where the orbital @ satisfi& hH_@= [-iv2 and H&tree-Fock
-Z/?l
(1) both the Hartree
+J(l)]@
= $+’
(2)
~ -
: hHF~.=[~BVZ-Z/r;+~(l)-~(2)]~ equations with the usual definitions.
YE;m
(3) .(
V&me
6, nuder
3
CHEMICAL PHYSICS LETTERS
.: --.
41)
&l);(2)
*)d~~;
= ( j-+(2) $-f(2)
The first-order [Ml)
= J;Pw
dr,)$(l)
1
El -+7(W(2)
- 2s] @l(rlY2012).=
.-q
.
equations are
+M2)
1 August 1970
-.
(4)
@(W(2)
and
1 [hHF(l) +hHF(2)- 2~1*10’1@1,] = LE1
.
where
(5)
1
-&+2J(1)-K(:)+2c7(2)-K(2,
$(l)@(2),
El = j-@2(1)L@2(2)drldr2 ‘12 is the first-order energy. In general, the functions +I and P1 depend cn several variables and eqs. (4) and (5) cannot & solved conveniently in this form. By expanding the wavefunctions as follows, q(y1y2Y2) %cosQ1i),
*l&2812)
= T - yly2
(6)
it is possible to reduce the first-order equation to an infinite set of uncoupled differential the functional coefficients q(rlY2). The 2th member of this set has the form
equations for
[-;(~+~)-~-=*~+~+~Yl,+~(;~)_2~]~z~ylY2)
2
1
2 2 +-< QJ‘- Yl+ 1 I ~&-lPlJ9-2)
= [ L91+&9+J(y2)] for the. Hartree case and [
22 _- 1- _+_ 2( JYf
a2 aY;
>
_.z_z ?1
t w Y2
1
f 249-l) + 2J(?-2) - 2E
+ E(Z+ 2y;
.2y;
(7)
>
lq(YlY2)
YE =[[Jq+J(q)+J(Q)] 610 --& q&-ljP&-$ +F&Y~~~)P~&I) f q(y2y1)~ls(~21
I
for the Hartree-Fock
case, where
&!(7-19.2) = -L 2z+ 1 .j-P1&3)
P-3)
y>
-$&1~3)
dy3
and r,:= max(r2r3) ;
r< =. min (Y2r3) .
:. Once the functional-coefficients -pres&on
are.fouud,
the second-order
energy can be evaIuated from the ex.-
_-
x
:-
._ with
an
identical expression
for: the Hartree
-
;
case. The. third-order.
“: qj-,=. &!I /IQ I&> -~‘.E&f~ j*‘l>.L .i&&l), __-.. _-. _ . .. . ‘. . _. .. _: ‘_ -: :,,i_:.-: ‘1’76 : __ .,, .: ,I’_ : _.<.- ._-: _:-: _,: , ,. . ._. -_ _.... .. ..;‘. .:. 1 _
-q@(q)
Y -,
_-
_.
+4y2)])
Qqy2) :
energy -is found’f&
‘- [
:
do dy2
,
(9)
._
”
(ida) : -. .: ., .-;. -._.. y:_._ .I..: _i_yy _-:,:_.:--:;._ : ;_, ~_~,,__,‘.~__..‘~___ I-‘._.: __. .‘.. -:-.__-. _- :_:. . . . -..._ ,_ ,-_...-_ ,‘L ..-. ; .:I ._ _..‘. .,.__-, .. ‘;.. . . ..^ ._ _,
I
..
._ x
Volume 6. number 3
CHEMICALPHYSICSLETTERS
which Ieads to the following
f August 1970
two forms, k
= Z F k ckU, Z’) &rlr2) ? ’
+-tree)
-&(y1~z)d~lcb-2 y>
(lob)
2 Eg(Ha.rtree-Fock)
= I F k ck(Z, I’) lul(rLr2) , 9
*
WlY2)
drl d72
?->
ww
3. NUMERICAL ANALYSIS As noted previously, the solutions to eqs. (4) and (5) were found with the MFD method. This involved replacing the derivatives by a difference approximation and solving the resulting set of linear equations. The calcultitions reported here were carried out with the fourth difference approximation,
The difference truncation error [the neglect of higher differences in deriving (II)]was removed by Richardson [4] extrapolation. In order that the method could be applied to the more diffuse pairs such as excited states and negative ions, the coordinates were transformed as follows: y1
=x;;
Y =x2 2 2’
On a grid in x1 and x2, the cutoff at which the functional coefficient is required to vanish can be taken sufficiently large without adversely affecting the grid spacing. The results presented in the next section were extrapolated from grids containing 20, 25, 30, and 35 strips with a cutoff at rL or 1~ = 9 au. 4. RESULTS AND DISCUSSION The zero-order solution was taken as Clementi’s [5] five-term expansion. This_ gives A?&= -1.83592 au and E 1 = -1.02576au; .The pareal wave contributions to the second-order energy for the Kartree and HsrtreeiFock calculations are compared’to the variational results of Byron and Joachain [S] in tabIe 1. The MFD results for the I _=0 wave are estimated to be above the -exact value by about 0.000002 au.. The higher partial waves are-more abcurate. .Part of the discrepancy bekeen the variational and numerical results can be.attributed to the use of different zero-order functions, Overall the agreement is quite ‘reasonable. In table 2, the total perturbation energies arecompared to those of Byron and Joachain [6,7] tid of Weiss and-Martin [2]; The calcujations in the thtid-and fourth-colu&s were carried out ‘-1with Hylleiaas trial functions -rather .tha.n by a pa.rtiali.wave ezpansion..~The disagreement between the -, ,. . . ._._ _I, _._:..., _,: -.: .- . -177 . . . .: ..’ ., -_ ._. -.
Volume 6. number 3
CHZMICAL
Comparison
PRYSICS LETTERS
of the partial-wave
1’;
ccdributions
to E2.
Hartree-Fock b
-E2 (0)
-
MFD
Rgrtree
Variztiond
-0.013498
1 August 1970 ;
2
a)
MFD
-0.01347
-0.018022-
:
Variational a) -0.01798
E2(1)
-0.018980
-0.01894
-0.024800
-0.02475
E2 (2)
-0.003193
-0.00317
-0.003676
-0.00365
E2(3)
-0.000932
-0.00092
-0.001014
-0.00100
E2(4) E2 (5)
-0.000360
-0.00035
-0.000381
-0.00038
-0.000116
-0.00016
-0.000172
-0.00017
E2 (6)
-O.Z)OOwG
-0.00008
-0.000088
-0.00009
E2h
-0.O30048
-0.000049
E2 N!
-0.000029
-0.000029
E2 (9,
-0.OOOO3.3
E2
-0.037305
, ‘_ i
~0.000019
I -0.03725
-0.048250
-0.04817
a) Ref. 161: Table 2 Comparison of the total perturbation energies MFD
Ref. [S]
Byron Ref. [7]
Weiss Ref. [2]
-0.04604
-0.04835
Hartr.?e E2 E3
-0.048250 OIOO731~
-0.04817 0.0071?
E
-2.902617
-2.90267
=2
-0.037311
-0.03725
E3 E
-0.003685 -2;902656
0.00713
0.00741
-2.90258
-2.90262
-0.03719
___
-0.00377
-0.00346
---
-2.90269
-2.90232
w-m
Hartree-Z’ock
MFD results and those of Weiss is due mainly to the neglect of higher partial waves. Byron and Joachain [S] have estimated the contribution to E2 from partial waves with I 3 ‘7 to be -0.00915 au. This would in,dicate at-least an error of -0.00006 au in the MFD value due ta truncation at Z = 9. The corrected value is -0.04331~au which compares- better with that of Weiss. The third-order energy also suffers-from the truncation. The total energies do not compare unfavorably wi:h the exact value [8] of -2.9037 au or the best CI value [9] of -2.9032 au. In order to characterize the first-order solution, several one-electron properties were computed and are presented in table 3. In parallel with the contributions .to E2, the Hartree-Fock solution has considerably smaller expectation values than the Hartree solution. The agreement is better for the higher partial waves. For the exact first-order wavefunction of a closed.-shell state, the.firsGorder’corrections to the expectation values, (@O~*@I>, are zero. The amount tr, whjch they do not.vanish is a measure of the accuracy of the wavefunction and the extrapolation procedure. The quantities (j-) and (y_2>are both. improved trj’the second-order correction; however, the expectation value of l/r is bronght into spme,what worse agreement.with the exact -mlue. .., The re@ts presented here have ahown.that the Hartree. a.ud Hartree-Fock pair equations can be SC&& accurately aud efficiently with the MFD method: Bi+use it&.not va+tio&il, this approach w&i be most useful when applied to excited states._ Calculations,are in progress on the its2 IS autoionizing states of 3he two-electron atom usiing zero-order solutions sjmilar to those iecently.published [lO];..The the ground_state apply without a:bdjficatiop to thesy configui&ong. ‘. _. ,.-( .. ., ._. .:,_ _:-f.. ::_ :. ,,:-: ,I,’ ‘. ..:_ _:‘‘I ; .’ ..^. I .. ,‘-.
Volume
6. number 3
CHEMICAL
Partial-wave
PHYSICS
analysis
LETTERS
Table 3 of one-electron
f August 1970
properties Hartree-Fock
Hartree P (l/r)
t
w
(l/r)
+->
~~
0
0.017601
0.013216
9.927072
0.009356
0.006283
0.012363
1
0.012350
0.012460
0.017951
0.006890
0.006066
0.007781
2
0.000747
o.oou875
0.001228
0.000539
0.000575
0.000772
3
0.000113
0.000141
0.000201
0.000092
0.000108
0.000149
4
0.000027
Q.OOOQ35
0.000050
0.000023
0.000029
o.oooo4c
5
0.000008
9.000011
0.000016
0.000006
0.000010
0.0000t4
6
0.000003
0.000004
0.000006
0.000003
0.000004
0.000005
0.030851
0.026745
0.045627
0.016923
0.013077
0.021129
.@% C%lFl*k3
-0.00006
-0.00001
-0.00001
-0.00001
-0.aaa04
<*fJ IfiI*fj
3.3748
1.8546
2.3696
3.3748
1.8546
2.3696
TOW
3.3693
1.8613
2.3895
3.3727
1.8572
2.3773
Exact a)
3.3766
1.8589
2.3970
3.3766
I.8589
2.3870
(*I
0.010776
1%)
a) ref.
0.00001
0.005637
[S].
REFERENCES 0. Sinanoj$lu, Advan. Chem. Phys. 14 (1968) 237: Proc. Roy. Sot. A260 (1961) 379. A. W. Weiss and J. B. Martin, Phys. Rev. 132 (l963) 2116. N.W. Winter, Ph. D. Thesis, California Institute of Technology (1970). L. Richardson and J. Gaunt, Trans. Roy. Sot. A226 (1927) 299. E. Clementi, IBM 5. Res. Develop. Suppl. 9 (1965) 2. F. W. Byron and C.J. Joachain. Phys. Rev. 157 (1967) 1. F. W-Byron and C. J. Joachain, Phys. Rev. 146 0966) 1. C. L. Pe’keris, Phys. Rev. 126 (1962) 1470. A. W.Weiss, Phys.Rev. 122 (1961) 1826. D. E. Ramaker, R. B. Stevenson and D.M. Schrader, J. Chem. Phys. 51 (1969) 5276.
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