- Email: [email protected]
On the relativistic corrections for H+2
VoIume 62, number
CHEMICAL
3
15 April 1979
PHYSICS LElTERS
ON THE RELATXVISTIC CORRECTIONS FOR H; J-W_ GONSALVES
zmd R-E_ MOSS
R+ceiwd t i December 1978
T& operator that ius been used to estimzttt;reIathistic corrections for the ground state of W; is discussed.It ij shol%n that to order mcz~4 no problems \:ith dixergcnt inte_rr;tfs arise_An alternathc operator. that has contputatiowi nduntases,
ir propord.
Tfie rquivaknce of the two operators is also demonstr.tted numeric&
Relativistic effects are important not only for mofecufes containing heavy atoms but also for light species such 3s H; for which Iii&-resoIutioti spectral data is avaifsble. Although full relativistic calculrttions for atoms have become cummon, relativistic cakul~tions for molecules are in their infzmcJt and for some time yet reIativistic corrections will be estimated from non-relativistic wavefunctions using perturbation theory. This approach is believed to be adequate for Ii&t molecules but the question arises xs to what is the most ztppropriate form for the non-relrttivistic approsimation COthe hamiItonian and what are the restrictions on its use. The importance of this question, esen for one-efectron problems, ~.IS been recognised for a tong time [I 1 but the f&t that no conclusive work has been done on it indicates how diffkuft the probIem is [?I_ Although this paper is concerned specificaIIIy with q, the conclusions are relevant to the more general problem. The refativistic wa\e equrttion for rut electron is the Dirac equation and for the electron in trg it may be written in terms of the Dirtic 1lamiItoni.m J&=
&XC +coL - p - I^e2/4zre&llrt
-f If+)_
0)
where the symbols have the usual meaning (see for eszt~np!e ref. (31) snd rI znd r, are the distances of the electron from nudei I and?. It is convenient to change to dimensionless coordinates in which lengths are measured in units of no, the Bohr radius, and to introduce a_ the fine structure constant. The Dirac hamiltonian (1) then becomes 534
-
+ I/r&-
HD =FFrc2~P’Qu’p-~Q7(f/rl
(3
L’nlike the non-relativistic problem [3], the Schriidinger equation, the relativistic problem for Hf cannot be solved exactly f5] _ There are ;Lnumber of methods f-or reducing the Dirac equation to nonrelativistic form (see for example refs. [2,31) bnt is each case the resulting hsmiftonhn may be expressed as a power series in Q
@ =(“-P).
E = -(f/r*
f I/Q7
(41
the components of abeing the Prtuli spin matrices. The Ieading term rxc2 is the electron rest energy and is omitted in the remainder of this paper, the term of order f&u2 corres onds to the non-relativistic 9 energy (note that t?- 5 p”), while the relativistic corrections that a-e of interest here are the expectation values of the o&_x’ terms_ These corrections have been estimated several times For the ground electronic state ofG using the exact non-relativistic wavefunction. Bishop [6] foilowed Luke et al. f?] in using the perturbation (in our units) t&x4
r-q [EQ •t 4hjR(?2
-J.?)]’
Volufne 62. number 3
CHEMICAL
where X = (r-1 + r,)/R and spheroidal coordinates, R is the zeroth-order energy units of n&c?_ In (5) Ku K, = [(1 -I-_: cr’E,-,)&
PHYSICS LETTERS
p = (r, - t-*)/R are prolate and E.
is the bond length
in atomic units. that is in
is a function
- p’) i- ?&i/R]
= I i-O(cl’)_
of A, p and E. ++Ka
-I (6)
Luke et al. [7] give no justification for their hamiltonian beyond a reference to Lowdin [8] _ In addition, they state that the approximation of replacing Kn by unity “is not vahd for the twocentre Coulomb potential with singularities at X = 1, p = i I”_ In other work on HS [9] the need for the factor K. is aIso emphasized iTdivergent resuIts are not to be obtained. Certainly divergences do arise in calculations to order nzc’cc4 and these divergences get progressively more serious for hi&her orders JlJO-_12] but in atoms no problems occur to order ntc-o14_ It 1s one of the purposes of this paper to show that this is also true for Hz and that K. can be replaced by unity in (5) without encountering any problems_ In addition_ the status of the operator (5) is discussed artd it is compared with other possible perturbation operators. In particular it is pointed out that one of the other forms is easier to use. The partitioning method of Lowdin [S] yields H= ,1,W(+KO
+ E).
(7)
where K=
[l -~&E--E-)1-I,
(8)
mc’( 1 + CU’E) being the exact relativistic ener_q. As already noted Luke et al. [7] do not describe the derivation of their hsmiItonian but, using the commutation relation [S] [&A’]
= $Yk[O.e]K
(9)
and the fact [S] that when K occurs at the estreme right of an eapression it may be replaced by (1 - $02K0K0), the hamiltonian (7) may be put in the form (3) with the relativistic perturbation ,,,,2,4
at tIlti
{-~e~li-02s
15 April 1979
- $e~[e, E]ir);
00)
stage the hamiltonian is exact. Similar manipuIations and substitution for 0 and E [eq. (4)] may then be used to obtain the approximate perturbation
- (rl jr: + r&)
Xp) + 0(nzc2a6)_
01)
The last tern1 represents spin-orbit coupling and may be omitted for Hz in its ground state since in that case its expectation value vanishes. When spheroidal coordinates are introduced and E in K is approximated by E. then (11) becomes (5). However, it is important to note that the retention of K in (11) is arbitrary and reference to (10) and ref. [9] shows that R’ cou!d have been kept instead_ On the other brand formal manipulations could be used to remove ii entirely from the terms of order IX-or ’ ’ in (11). Presumably K was retained by Luke et al. because of the danger of a divergent expectation value_ In fact detailed considerations of the integral involved in the expectation value of the second term m (11) shows that no divergence occurs when A’ is replaced by unity [ 131; this assertion has also been confirmed numerically (see !citer)_ We note in passing that (11) is not hermitean; this can be remedied by renormalisation (see for example reFs_ [2_3.10.14]) but thiihas not been done for H;. One of the other methods of reducing the Dirsc hamiltoninn to non-relativistic form is that due to FoIdy and Wouthuysen (see for exampie refs. [2.3] ) in which e unitary transformation is used to uncouple positike and negative energy coml-onents of the Dirac wavefunction The resulting non-relatrvistic hamrltonian is thus hermitean and in addition no factors like K appear so that each term has a definite order of magnitude. is mc’ff4
The standard result for the perturbation
c--i 0” - b [O, [O_E] ] ) + O(mc2a6):
(12)
this is to be compared with (IO) when K has been replaced by one_ Since we are only interested in tile espectation value, the non-relativistic Scbriidinger equation
($0’ +E)s,
=EoGo
(13)
can be used to replace 0” by 2(Eo - E) when it operates on the zeroth-order wavefunction. Thus 0’ may be replaced by
535
Volume 62, numer 3
15 April 1979
CIZESIICAL PHYSICS LETTERS
+-iJ - & - 2[0’,&
(14)
this procedure w3s in f3ct used earlier to obtain (i I). The perturbation mc’oif
f-$(q)
(12) then becomes - E)? +_d [UQ
-i[O,
[O,EJ 1)
+ (s(mc’ff6) and substitution ?&a~
-I-6 (nrc’o16)_
(15)
Tile fact that a term in the perturbation may be replaced by another is perhaps not surprising since the non-unitary transformation, which cannot affect the eigenvalues of the hamiltonian, can only add a term which has zero expectation vaIue so that, since
for 0 3nd E eventualiy yields
t-4 (k-0 + I [J-, f- I /@
[O’.E]
= -4iT[S(q)
- WI/~: If R is repI3Lpd by unity. then it is possible to relate the perturbation (12) to the perturbation (IO) [or (16) to (I I)] by performing a further transforn~tion on the hamiltoni3n obtained using the FoldyWouthuysen tr3nsformation_ This tr3nsform3tion is not unit3ry and introduces non-hermitean terms into the tmusformed hamiItonian but this does not affect the eigenv3Iues of the hamiItonian_ In the FoldyWouthuysen transformation WD is tr3nsformed to exp(isyr, exp(--S) where S is 3n hermitetln operator. but if S is augmented by the non-hermitean oper3tor --iu’X_~U’, where k is 3n adjustable constant, then the perturbation (12) becomes ,,lAJ
{-$?J
+ 0(nrc%@
- g?.[O.E]] )_
(18)
*a[o’.e]) (17)
UJ choosing k = -f we obtain (10) with K replaced h>- unity. fIowr\er. any choice of k is possibie and comparison of ( 16) with (1 I ) shows tin: it should be possible to eliminate rhe term imoking (rt/< +r&) - p. ~htch gives rise to the complicated differential operator in (5). entireIy in f3vour of delta-function operators: this is desir3bIe since the calculation of :hc expectation value of a delta-function operator only involves sampling the wavefunction 3t the nucleus .md does not involve numerical integration. The appropriate choice is k = -$ and the perturbatlon becomes
+6(r*)]
+f&
l
P,
(19)
one mi& espect -$i(rt /G +t-&) - p and $ ~[?j(r~) + S(r?)] to be equivalent-in the sense that they have the same expectation values; 3s we will see this has been confirmed numerically. if the spin-orbit coupling term is omitted and spheroidal coordinates introduced, the perturbation corresponding to the reIativistic corrections for the ground stare of Hz becomes J,Z&X~ {-i
[E.
f 4A/R(X’
- P’)]’
+ 6 (m2,6);
(20)
this is to be compared with (5) These results have been confirmed numerically and the details 3re given in table I which compares the Hz ground-state e.xpectation v3hxes B = <[4K,,/R’(A’ x [(A’ +&(A’
- p’)“] -
I)a/ax
- Z&I
- &a/a& (21)
3nd
Volume 62. number 3
CHEMICAL
PHYSICS
Table 1 Comparison of B (21) and D (22) for the gound electronic state of H; R (co)
-
0.0 =) 02 0.4 0.6 0.8 1-O 1.1 1.4 1.6 1-S 2.0 1.1 2.4 2.6 2.6 3.0 4.0 5.0 6-O 7.0 8.0 9.0 10.0 Gea)
B
D
s-0 5.02943 1 3287310 1.311681 I_730281 1.361308 1.114601 0.943610 0.818792 0.727297 0.658381 0.605738 0.565 149 0.533709 0.509358 0.490597 o-449351 0.451662 0.466471 O-48023 I 0.489303 0.494396 OA97030 0.5 -__
8.0 5.032238 3.288998 23 I2824 1.731114 1.361963 1.115134 0.94309 1 0.819184 0.727643 0.658695 0.606026 0.565418 0.533963 0.509603 0.490830 0.449569 0.451879 0.466695 0.480161 0.489540 0.49-t633 0.497169 0.5
a) United and separated atom lima w~\efunctions. Although
Bishop
[6]
cakulated
so
We are most grateful to Professor D.M. Bishop of the University of Ottawa for generously providing the values of B given in the table and for recalculating B with K, = I _ One of us (J.W.G.) thanks the British Council and the C_S_l.R_ (South Africa) for support_ References [ 11 J-E_ Harriman. Preprint 127. Quantum Chemistry
(23)
that
(24)
In table 1, B and D agree to about 3 significant However,
of a factor such as
in (5) to avoId divergences is unnecessary; if the factor R, is retained some additional contributions of order n&o6 and higher are included in the calculated values but this is not significant as there are other contributions which are omitted. In addition, it is computationally advantageous to employ the operator (20) which involves delta functions rsther tlnm (5) involving differential operators, whether K. is set to unity or not.
only quotes the total relativistic
D=&(R).
independent of R. being about 1.0005 [ 161, but there appears to be no simple explanation of this feature_) It has been shown that although divergent integmls may be encountered in the calculation of relativistic ’ 6 , no such difficulty arises corrections to order nzc-Q! to order NIC%Y~ and the retention
using atomic
= (Sr#)P(R),
15 Aprd 1979
K,
correction he calculated B x an intermediate step and has made the vaiues quoted in table 1 available to us. The values of D in table 1 were cakulated from the tabulated S,(R) of Bishop and Cheung [ 151; their S,(R) are related to p(R), the absolute value of the electron density at the nucIeus, by S#)
LETTERS
figures.
it should be noted that B includes the
factor K. and it is only when K. is replaced by unity in (3 1) that the equality of B and D is expected_ Bishop [ 161 has repeated his calculation of B, but with K. = I, and obtains agreement with D to 6 significant figures. (7he ratio of D to B in the table is almost
Group, Uppszla Unirersiry, Sweden (1964), unpublished_ 111 E. de Vries. Fortschr- Physik 1s (1970) 149. [3] R.E. Moss, Adxmced molecule quantum mechrtnicstm innoduction to rehrivistic quantum mechanics and the quantum theory of mdmtmn (Chapman and Ii&, London, 1973). [4] C-X. Coulson and A. Joseph, Intern. J- Quantum Chem_ 1 (1967) 337. [5] B. ifuller and 1%‘.Greiner, Z. Naturforsch. A 31 (1976) I. [6] DX. Bishop, J_Chem.Phys. 66 (1977) 3541. [7] SK. Luke, G. ilunter, R-P_ .\IcCxhran .md Si. Cohen, J_ Cbem. Phys_ 50 (1969) 1614_ 181 P--O. Lo\\dm. J. Mol. Spectr!. 14 (196-l) 131_ is1 E Xl_ Roberts, 11.R. Foster snd F.F. Selii, J. Chem. Phys. 37 (1962) 485. [lOI R_ .\Ic\Veeny snd B-T. Sutcliffc. Methods of molecular quantum mechxics (Xcadcmic Press, NW York, 1969) app. 4. 1111 Xl. Douglas, Phqs. Rev. -L\I1 (1975) 1527. 1121 I.D. Morrison and R.E. Moss, to be submitted for publication. J.W. Consalves. unpublished_ A-1. _tihiezer .md V.B. Berestetskii. Quantum electrodynamics (Interscience, New York. 1965). D-M. Bishop and LX_ Cheung. J. Phys- B 11 (1978) 3133. D.&l. Bishop, private communication.
537
CHEMICAL
3
15 April 1979
PHYSICS LElTERS
ON THE RELATXVISTIC CORRECTIONS FOR H; J-W_ GONSALVES
zmd R-E_ MOSS
R+ceiwd t i December 1978
T& operator that ius been used to estimzttt;reIathistic corrections for the ground state of W; is discussed.It ij shol%n that to order mcz~4 no problems \:ith dixergcnt inte_rr;tfs arise_An alternathc operator. that has contputatiowi nduntases,
ir propord.
Tfie rquivaknce of the two operators is also demonstr.tted numeric&
Relativistic effects are important not only for mofecufes containing heavy atoms but also for light species such 3s H; for which Iii&-resoIutioti spectral data is avaifsble. Although full relativistic calculrttions for atoms have become cummon, relativistic cakul~tions for molecules are in their infzmcJt and for some time yet reIativistic corrections will be estimated from non-relativistic wavefunctions using perturbation theory. This approach is believed to be adequate for Ii&t molecules but the question arises xs to what is the most ztppropriate form for the non-relrttivistic approsimation COthe hamiItonian and what are the restrictions on its use. The importance of this question, esen for one-efectron problems, ~.IS been recognised for a tong time [I 1 but the f&t that no conclusive work has been done on it indicates how diffkuft the probIem is [?I_ Although this paper is concerned specificaIIIy with q, the conclusions are relevant to the more general problem. The refativistic wa\e equrttion for rut electron is the Dirac equation and for the electron in trg it may be written in terms of the Dirtic 1lamiItoni.m J&=
&XC +coL - p - I^e2/4zre&llrt
-f If+)_
0)
where the symbols have the usual meaning (see for eszt~np!e ref. (31) snd rI znd r, are the distances of the electron from nudei I and?. It is convenient to change to dimensionless coordinates in which lengths are measured in units of no, the Bohr radius, and to introduce a_ the fine structure constant. The Dirac hamiltonian (1) then becomes 534
-
+ I/r&-
HD =FFrc2~P’Qu’p-~Q7(f/rl
(3
L’nlike the non-relativistic problem [3], the Schriidinger equation, the relativistic problem for Hf cannot be solved exactly f5] _ There are ;Lnumber of methods f-or reducing the Dirac equation to nonrelativistic form (see for example refs. [2,31) bnt is each case the resulting hsmiftonhn may be expressed as a power series in Q
@ =(“-P).
E = -(f/r*
f I/Q7
(41
the components of abeing the Prtuli spin matrices. The Ieading term rxc2 is the electron rest energy and is omitted in the remainder of this paper, the term of order f&u2 corres onds to the non-relativistic 9 energy (note that t?- 5 p”), while the relativistic corrections that a-e of interest here are the expectation values of the o&_x’ terms_ These corrections have been estimated several times For the ground electronic state ofG using the exact non-relativistic wavefunction. Bishop [6] foilowed Luke et al. f?] in using the perturbation (in our units) t&x4
r-q [EQ •t 4hjR(?2
-J.?)]’
Volufne 62. number 3
CHEMICAL
where X = (r-1 + r,)/R and spheroidal coordinates, R is the zeroth-order energy units of n&c?_ In (5) Ku K, = [(1 -I-_: cr’E,-,)&
PHYSICS LETTERS
p = (r, - t-*)/R are prolate and E.
is the bond length
in atomic units. that is in
is a function
- p’) i- ?&i/R]
= I i-O(cl’)_
of A, p and E. ++Ka
-I (6)
Luke et al. [7] give no justification for their hamiltonian beyond a reference to Lowdin [8] _ In addition, they state that the approximation of replacing Kn by unity “is not vahd for the twocentre Coulomb potential with singularities at X = 1, p = i I”_ In other work on HS [9] the need for the factor K. is aIso emphasized iTdivergent resuIts are not to be obtained. Certainly divergences do arise in calculations to order nzc’cc4 and these divergences get progressively more serious for hi&her orders JlJO-_12] but in atoms no problems occur to order ntc-o14_ It 1s one of the purposes of this paper to show that this is also true for Hz and that K. can be replaced by unity in (5) without encountering any problems_ In addition_ the status of the operator (5) is discussed artd it is compared with other possible perturbation operators. In particular it is pointed out that one of the other forms is easier to use. The partitioning method of Lowdin [S] yields H= ,1,W(+KO
+ E).
(7)
where K=
[l -~&E--E-)1-I,
(8)
mc’( 1 + CU’E) being the exact relativistic ener_q. As already noted Luke et al. [7] do not describe the derivation of their hsmiItonian but, using the commutation relation [S] [&A’]
= $Yk[O.e]K
(9)
and the fact [S] that when K occurs at the estreme right of an eapression it may be replaced by (1 - $02K0K0), the hamiltonian (7) may be put in the form (3) with the relativistic perturbation ,,,,2,4
at tIlti
{-~e~li-02s
15 April 1979
- $e~[e, E]ir);
00)
stage the hamiltonian is exact. Similar manipuIations and substitution for 0 and E [eq. (4)] may then be used to obtain the approximate perturbation
- (rl jr: + r&)
Xp) + 0(nzc2a6)_
01)
The last tern1 represents spin-orbit coupling and may be omitted for Hz in its ground state since in that case its expectation value vanishes. When spheroidal coordinates are introduced and E in K is approximated by E. then (11) becomes (5). However, it is important to note that the retention of K in (11) is arbitrary and reference to (10) and ref. [9] shows that R’ cou!d have been kept instead_ On the other brand formal manipulations could be used to remove ii entirely from the terms of order IX-or ’ ’ in (11). Presumably K was retained by Luke et al. because of the danger of a divergent expectation value_ In fact detailed considerations of the integral involved in the expectation value of the second term m (11) shows that no divergence occurs when A’ is replaced by unity [ 131; this assertion has also been confirmed numerically (see !citer)_ We note in passing that (11) is not hermitean; this can be remedied by renormalisation (see for example reFs_ [2_3.10.14]) but thiihas not been done for H;. One of the other methods of reducing the Dirsc hamiltoninn to non-relativistic form is that due to FoIdy and Wouthuysen (see for exampie refs. [2.3] ) in which e unitary transformation is used to uncouple positike and negative energy coml-onents of the Dirac wavefunction The resulting non-relatrvistic hamrltonian is thus hermitean and in addition no factors like K appear so that each term has a definite order of magnitude. is mc’ff4
The standard result for the perturbation
c--i 0” - b [O, [O_E] ] ) + O(mc2a6):
(12)
this is to be compared with (IO) when K has been replaced by one_ Since we are only interested in tile espectation value, the non-relativistic Scbriidinger equation
($0’ +E)s,
=EoGo
(13)
can be used to replace 0” by 2(Eo - E) when it operates on the zeroth-order wavefunction. Thus 0’ may be replaced by
535
Volume 62, numer 3
15 April 1979
CIZESIICAL PHYSICS LETTERS
+-iJ - & - 2[0’,&
(14)
this procedure w3s in f3ct used earlier to obtain (i I). The perturbation mc’oif
f-$(q)
(12) then becomes - E)? +_d [UQ
-i[O,
[O,EJ 1)
+ (s(mc’ff6) and substitution ?&a~
-I-6 (nrc’o16)_
(15)
Tile fact that a term in the perturbation may be replaced by another is perhaps not surprising since the non-unitary transformation, which cannot affect the eigenvalues of the hamiltonian, can only add a term which has zero expectation vaIue so that, since
for 0 3nd E eventualiy yields
t-4 (k-0 + I [J-, f- I /@
[O’.E]
= -4iT[S(q)
- WI/~: If R is repI3Lpd by unity. then it is possible to relate the perturbation (12) to the perturbation (IO) [or (16) to (I I)] by performing a further transforn~tion on the hamiltoni3n obtained using the FoldyWouthuysen tr3nsformation_ This tr3nsform3tion is not unit3ry and introduces non-hermitean terms into the tmusformed hamiItonian but this does not affect the eigenv3Iues of the hamiItonian_ In the FoldyWouthuysen transformation WD is tr3nsformed to exp(isyr, exp(--S) where S is 3n hermitetln operator. but if S is augmented by the non-hermitean oper3tor --iu’X_~U’, where k is 3n adjustable constant, then the perturbation (12) becomes ,,lAJ
{-$?J
+ 0(nrc%@
- g?.[O.E]] )_
(18)
*a[o’.e]) (17)
UJ choosing k = -f we obtain (10) with K replaced h>- unity. fIowr\er. any choice of k is possibie and comparison of ( 16) with (1 I ) shows tin: it should be possible to eliminate rhe term imoking (rt/< +r&) - p. ~htch gives rise to the complicated differential operator in (5). entireIy in f3vour of delta-function operators: this is desir3bIe since the calculation of :hc expectation value of a delta-function operator only involves sampling the wavefunction 3t the nucleus .md does not involve numerical integration. The appropriate choice is k = -$ and the perturbatlon becomes
+6(r*)]
+f&
l
P,
(19)
one mi& espect -$i(rt /G +t-&) - p and $ ~[?j(r~) + S(r?)] to be equivalent-in the sense that they have the same expectation values; 3s we will see this has been confirmed numerically. if the spin-orbit coupling term is omitted and spheroidal coordinates introduced, the perturbation corresponding to the reIativistic corrections for the ground stare of Hz becomes J,Z&X~ {-i
[E.
f 4A/R(X’
- P’)]’
+ 6 (m2,6);
(20)
this is to be compared with (5) These results have been confirmed numerically and the details 3re given in table I which compares the Hz ground-state e.xpectation v3hxes B = <[4K,,/R’(A’ x [(A’ +&(A’
- p’)“] -
I)a/ax
- Z&I
- &a/a& (21)
3nd
Volume 62. number 3
CHEMICAL
PHYSICS
Table 1 Comparison of B (21) and D (22) for the gound electronic state of H; R (co)
-
0.0 =) 02 0.4 0.6 0.8 1-O 1.1 1.4 1.6 1-S 2.0 1.1 2.4 2.6 2.6 3.0 4.0 5.0 6-O 7.0 8.0 9.0 10.0 Gea)
B
D
s-0 5.02943 1 3287310 1.311681 I_730281 1.361308 1.114601 0.943610 0.818792 0.727297 0.658381 0.605738 0.565 149 0.533709 0.509358 0.490597 o-449351 0.451662 0.466471 O-48023 I 0.489303 0.494396 OA97030 0.5 -__
8.0 5.032238 3.288998 23 I2824 1.731114 1.361963 1.115134 0.94309 1 0.819184 0.727643 0.658695 0.606026 0.565418 0.533963 0.509603 0.490830 0.449569 0.451879 0.466695 0.480161 0.489540 0.49-t633 0.497169 0.5
a) United and separated atom lima w~\efunctions. Although
Bishop
[6]
cakulated
so
We are most grateful to Professor D.M. Bishop of the University of Ottawa for generously providing the values of B given in the table and for recalculating B with K, = I _ One of us (J.W.G.) thanks the British Council and the C_S_l.R_ (South Africa) for support_ References [ 11 J-E_ Harriman. Preprint 127. Quantum Chemistry
(23)
that
(24)
In table 1, B and D agree to about 3 significant However,
of a factor such as
in (5) to avoId divergences is unnecessary; if the factor R, is retained some additional contributions of order n&o6 and higher are included in the calculated values but this is not significant as there are other contributions which are omitted. In addition, it is computationally advantageous to employ the operator (20) which involves delta functions rsther tlnm (5) involving differential operators, whether K. is set to unity or not.
only quotes the total relativistic
D=&(R).
independent of R. being about 1.0005 [ 161, but there appears to be no simple explanation of this feature_) It has been shown that although divergent integmls may be encountered in the calculation of relativistic ’ 6 , no such difficulty arises corrections to order nzc-Q! to order NIC%Y~ and the retention
using atomic
= (Sr#)P(R),
15 Aprd 1979
K,
correction he calculated B x an intermediate step and has made the vaiues quoted in table 1 available to us. The values of D in table 1 were cakulated from the tabulated S,(R) of Bishop and Cheung [ 151; their S,(R) are related to p(R), the absolute value of the electron density at the nucIeus, by S#)
LETTERS
figures.
it should be noted that B includes the
factor K. and it is only when K. is replaced by unity in (3 1) that the equality of B and D is expected_ Bishop [ 161 has repeated his calculation of B, but with K. = I, and obtains agreement with D to 6 significant figures. (7he ratio of D to B in the table is almost
Group, Uppszla Unirersiry, Sweden (1964), unpublished_ 111 E. de Vries. Fortschr- Physik 1s (1970) 149. [3] R.E. Moss, Adxmced molecule quantum mechrtnicstm innoduction to rehrivistic quantum mechanics and the quantum theory of mdmtmn (Chapman and Ii&, London, 1973). [4] C-X. Coulson and A. Joseph, Intern. J- Quantum Chem_ 1 (1967) 337. [5] B. ifuller and 1%‘.Greiner, Z. Naturforsch. A 31 (1976) I. [6] DX. Bishop, J_Chem.Phys. 66 (1977) 3541. [7] SK. Luke, G. ilunter, R-P_ .\IcCxhran .md Si. Cohen, J_ Cbem. Phys_ 50 (1969) 1614_ 181 P--O. Lo\\dm. J. Mol. Spectr!. 14 (196-l) 131_ is1 E Xl_ Roberts, 11.R. Foster snd F.F. Selii, J. Chem. Phys. 37 (1962) 485. [lOI R_ .\Ic\Veeny snd B-T. Sutcliffc. Methods of molecular quantum mechxics (Xcadcmic Press, NW York, 1969) app. 4. 1111 Xl. Douglas, Phqs. Rev. -L\I1 (1975) 1527. 1121 I.D. Morrison and R.E. Moss, to be submitted for publication. J.W. Consalves. unpublished_ A-1. _tihiezer .md V.B. Berestetskii. Quantum electrodynamics (Interscience, New York. 1965). D-M. Bishop and LX_ Cheung. J. Phys- B 11 (1978) 3133. D.&l. Bishop, private communication.
537