- Email: [email protected]
Pseudopotential calculations of alkali interactions
Volume 7; nnmber 5 .:.:-.
CHEMICAL PHYSICS LETTERS
1 December
:
-:.
<. .
197,
. ,,
‘. ,:.
,F’
CALCULATIONS
.PSEUD,OPOTENTTAL -
OF
ALKALI
INTERACTIONS
J. N. BARDSLEY * Physics Deparlment, University of Pillsbargh. Pittsburgh. Pennsylvania 15213. VSA
. . .._
Received
6 October
1970
Several forms of pscudopotentinl arc presented for the valence electron in lithium and sodium. tentials are used to cakulate potential energy curves for Liz and Lip. Comparison with ab initio confirms the usefulness of the pseudopotentiai technique.
Many properties
of alkali
ly on the behaviour
atoms
of the valence
depend
main-
electron_
How-
ever, for ab initio calculations of the motion of the valence electron one needs to know the wavefunctions of all the core electrons. This is because the nuclear charge is partially screened by the surrounding core electrons, and because the valence electron wavefunction must be orthogonal to the core functions. This requirement of orthogonality causes considerable complication in the calculation of inter-atomic potentials. Formally the orthogonality requirement can be avoided by the introduction of a repulsive pseudopotential which is added to the potential experienced by the valence electron [ 11. The use of pseudopotentials is now a standard technique in solid state theory [2] and is becoming popular in atomic and mol_ecular physics. The ab initio calculation of pseudopotentials has been pioneered by Szasz and McGinn [3]. An alternative approach is to construct the pseudopotential empirically (e.g. as in refs. [4] and [5]). Further references and a description of the techniques can be found in the review by Weeks et al. [6]. In this
paper
the
empirical
approach
is illus-
trated and some preliminary results presented on the potential curves for Lii and Li2. The effective potential experienced by the valence electron in an alkali is expressed in the form 1 ‘Yd Veff(r) = -7 2
% 2(r2 + d2)3 where
.
c!d and try9 are
the
dipole
+ VSR(r) , and quadrupoie
(1)
These pocalculations
polarizabilities of the core and d is a cut-off parameter which is roughly equal to the radiuz of the core. VsR( r) is a short range potential which is small outside the core of the alkali; it is given a simple form with parameters which are chosen so that the eigenvalues of the one Vef (r) electron hamiltonian with the potential reproduce closely the observed alkali spec i rum. A significant difference between Ve;erf(r) and the Hartree-Fock potential for the aikali is that the former has no eigenvalues corresponding to the core orbitals. Thus for lithium the lowest s-wave eigenvalue of Veff( r) is equal to the en&gy of the valence electron in the 2s state. The purpose of the effective potential is both to represent the partially screened attraction of th nucleus and to simulate the influence of the Pauli principle on the valence electron. Becaus’ of the latter role Veff(r) depends strongly on th angular momentum of the valence electron. For example in lithium an s-wave electron is effectively repelled from the nucleus because of the requirement of orthogonality to the 1s function of the core, whereas for any non-zero angular momentum the Pauli principle leads to no constraint on the wavefunction. Thus we write
In general
for the partial wave I, the potential if there are core electrons wit 2H angular momentum I, and is attractive otherwise. The effective potential defined by eq. (1) is
Vz (r) is repulsive
* Most of this work wss cnrried
out :tt the University of TCXIS at Austin. The author is grntcful to that univcrsitv for the provision of free computer time. 51
Volume 7. number
CHEMICAL PHYSICS LETTERS
5
C+tlatio_n$ of the form
correct asymptotically to order l/r6 if ihe core polarizabilities are known and if d is chosen correctly. Thus if we integrate the one particle Schrodinger equation with this potential inwards from 00using the correct energy, the wavefunction will have the proper form outside the core. The normalization may be incorrect. However, there is a theorem in Quantum Defect Theory [7] that relates the normalization of a valence electron wavefimction to the rate of change of quantum defect with energy as one moves through a Rydberg series. Thus if one ensures that the effective potential leads to good energies for all the members of each Rydberg series then one obtains accurate approximations to the valence wavefunctions outside the core. This can be accomplished by introducing two (or more) variable parameters into each radial potential v;R(1-).
:
State BS
Gs
Ir~cl
fitting
for sodium.
GP GP
_
5rI
Gd
=-Ar&xp,kS~
VP) .
1
series the ?a=3nnd n=4 levels
In each
Effective principzt Esponcnt in1 -_--_. --. --3.64743 4.64930
Spectroscopic _______________._-..-._---3.64735 4.tiQS18
y&(r)
1970
(2) Several choices of_integers have been tried forb and q; a simple exponential (p = 0; q = l), a Hellmann or ,Yukawa potential (#= -1; q = l), and a modified gaussian form (p = 2; q = 2). The fitting of the exponential energy levels is illustrated in table 1 which applies to sodium. In each case the parameters Al and 51 are chosen so th’_t the lowest two levels of each series are reproduced exactly (spin-orbit effects are ingnored). The effective principle quantum numbers of the next two levels of each series are given in table 1. The experimental spectrum. can be reproduced with high accuracy (0.1% in ionization or excitation energies) with each form and thus the choice of the type of potential should be made on other grounds. A check of the empirical approach
Table
Spcctml
.’ :I December
1’ have been m,adeIvith potenti,als i
hsve
been
fitted
csnctl!
quantum numhers Hcllmnnn
Modified gnussinn
3.64743 4.ci4S30
3.64739 4.64924 4.13837
..-
4.13838 5.1407’3
4.13869
4.13869
5.14111
5.14113
5.14077
4.98G79 5 . 9dG”S _
4.98680 5.98G30
4.98GSlJ 5.98630
4.S8681 5.98G31
_____-.__-__
_.____ ____.-.__._._ _______ ___________________._----_-. -___-------
Table 2 t~seudopotentinl pnrnmeters __ _.__ ----..-.__----
--_-_________ 1
Espnncnt
in1
gaussinn
Cl
AI
52
AI
Cl
56.5046
3.61625
26.7618
2.R9636
15.7672
2.04616
1
-5.lSIlZO
3.JtiSW
-1.854Gl
Z.(i756(i
-1.86840
2.1ti726
> 2
-1.00861
3.10241
2.G1247
-0.025134
0.86045
----
ad=0.1325
-.--.-
--
1
_-_-_-_--__--..-
aq=o.112 --------_--_
I%eudtrpotcntinl
d=O.75 --_
Tnblc 3 pnrnmetcrs for sodium -._ _-________
4.53358
Al ---.--__-___
___-_--A-------
lti4.89G
Modified Cl
AI
3.85772
28.8425
39.7714
2.92164
2G.3684
i.40297
-3.33147
2.19462
-2.78823
1.79360
cYd=o.945
-_-----
ticlimnnn (1 ---_
--.--.-.-_-------0 317.512 1
O.G9607
Esponentinl AI
22
518
Modified
0
---_
~.----_
-.
_____-_----____~-
Hcllmann
Al
___
for lithium
act = 1.623 _-_-___--__
gsussinn _-_-
51 2.16472
3.00436
1XJDOl
--0.109$9
0.55401
d=l.l --+-
______------
--
Vo[rime ii. number 5
CHEMICAL
PHYSICS LETTERS
that the result of any calculation should not depend stror&y on the form of pseudopotential used. The potential parameters for Li and Na are given in tables 2 and 3. .‘ For‘calcdlating the interaction of an alkali ion with a neutral alkali we consider the system ABf and construct effective potentials Vsff( r) and Vzff (r). The effective hamiltonian is then
1 Dcccmbcr
ii
II= -+v;+
VA eff( la ) + r’gff( b) ‘
+ Vc (R) v
(3)
represents the interaction bein which V,(R) tween the two cores separated by the vector R; ra and rh are the vectors between the valence electron and the two nuclei. In the calculations reported here Vc(R) has been taken to he l/R, neglecting core-core polarization and overlnpping of the two cores. At each separation R the electronic energy is calculated by the variational method with a trial wavefunction of the form *(I-) = c 4,1 &#-a) nz The basic atomic of the equation {-sv;
functions
+ c b,, +.yb) n ti$l(?;)
(4)
are solutions
+ hp$ff(*-a) - CA} 6$,(ra)
=9
(5)
and the @E[(rb) are defined in an analogous manner. The energy CA is taken to be the ground state energy of A. The scaling parameter X is treated as an eigenvalue for the generation of the basic functions. Atomic functions defined in this way form a complete set without any continuum. This method has been applied t&the calculation of the lowest 2Zi and 2s; stat& of Lii . The results using six expansion functions for each atom are given in table 4. The dissociation energy of 1.23 eV and the equilibrium nuclear separation of 5.9 au are close to the predictions of ab initio calculations [8,91. The technique can easily be extended to systems with several valence electrons. The results for the lowest I>* and 3X, states of Liz are also shown in tableif: The accuracy of the calculation is limited by the number of two electron-integrals that can be computed at a reasonabIe cost. Using a 13 configuration wavefunction the dissociation energy is calculated to be 0.89 eV, whereas the experimental energy is 1.05 eV [lo] and an ab initio calculation gives 0.99 eV [ 111. The equilibrium separation is close to the experimental value of 5.05 au. Before a final
Vdencc
R
Tal~lc 4 cncrgics for 12, :fnd Lip. tics nrc in :llomic units
electron
Li 2?x,
1970
All qwrnti-
Liz 2,.
R
-‘II
1x
R
3s .‘,I
6
-0.23SB
-0.1105
6.5
-0.2429
-0.1246
5.5
-0.4278
G
-0.2435
-0.1366
ti
-0.4241
-0.3847
6.5
-0.2421
-O.146!1
8
-~J.JlJ(il
-0.3!J.U
-0.2305
-0.1557
10
-0.3990
-0_3!Ki5
-0.23(i3
-0.1633
12
-0.3973
-0.3969
-I).1 OY2
-ll.l9S2
00
JJ.39ti3
-u.39ti3
7 ‘i..i
00
5
-0A200
-0.3790
judgement can be made on the technique the numerical techniques must be improved so that more configurations can be included. One problem that should be investigated by this method is the range of applicability of asymptotic forms of the long range forces between atoms. The calculations on Li2 show that the effects of overlap of electron orbit&s are important out to separations of 15 au which is three times the equilibrium separation_ Kutzelnigg [51 has shown that it is at similar separations that the Rs8 term in the multipole expansion becomes small in comparison with the pure van deriSVaals attraction, These facts may explain th’Z large difference between calculated van der Waals coefficients for alkalis and those
derived from scattering data [ 121. A different use of empirical potentials
for alkali valence electrons has been reported by Dalgarno et al. [13]. Their empirical potential does not simulate the effect of the Pauli principll momentum. and thus is not dependent on an@ar The eigenfunctions of this potential approximate both core and valence orbitnls, and the problem of orthogonality of core and valence functions remains. An advantage of their approach is that the valence functions have the correct number of nodes and thus may be better approximations inside the core region. They also treat the corecore interaction more carefully than has been
done here.
In conclusion it is our belief that empirical pseudopotential method can be used to obtain accurate potential curves but further work is needed to demonstrate this thesis. In choosing an empirical potential an explicit f-dependence should be introduced and at least two adjustable parameters included in each component. There are also many possible applications of the meth51!
Volume
7. .number
CHEMICALPHYSIC!S.~TTE~
5
__-. ‘:
_..
od to electron
and
photon
collisions
with
atoms;
as shown by the photoionization calculations of Smith and LaE!ahn [14].
REFERENCES [lj J. G. Phillips and L. Kleinman. Phys. Rev. 116 (19.59) 287; 118 (1960) 1351. I21 Pscudcmotcntinls in the theory of . W. A. Harrison. metals (Benjamin. Now-York. 1966). [31 I+Szasz and 0. McGinn. J. Chem. Phys. 45,(1966) 2898: 47 (1967) 3495: 43 (1969) 2997: L. R. Kahn and W. A.Goddard Ill. Chem. Phgs. Letters 2 (1968) 667. [4J L.Sznsz and G. BIcGinn. J. Chem. Phys. 42 (1965) 2363; J. Csllnwvny and P.S. Lnghos. Phys. Rev. 187 (1969) 192: G.Simons and A. Mazziotti. J. Chem. Phys. 52 (1970) 2-149. I
520
_-I J.Kooh.i:nd W. Kutzelni‘ggi Acta Ploys,.Acad. Sci:. ++%27. (19!?) 323: . -.. .. W.Kutze!niggi”Che,mr P.hys, Letters, 4_.(1970) 435, E. S. Giulaco and R:+gge’ri. Nuovo Cimcnto ~-61 : .’ -: .: (1969) 53. A’.Rice; ‘Ad&. Chem. 151J, D. Weeks. A’.H&i&d$ Phys. 16 (19.69) 283. .;:. ;’ iii &I_J. Seaton.. M9qthl.v Notices Roy: ‘&t&n. Soo. 118 (1958) 5f+.,-_: : ’ .’ J. Chemj,P_his_’ i $35). 9. ; Gl H;M. Jim&. phys. Rev. 186 191C. R. Fischer and P. J.Kommey. (1969) 272. -: [IO] 6. He&erg. Sptictra’of diatomic molecules. 2nd Ed, (Van Nostrand, Princeton. 1950) p. 546, Ill] G;Das. J. Chem. phys. 46 .(lSq 1568. 1121 A. Dalgarno and W. D.Dwison. A&an. A’toti. +. Phrs. 1 (1966) 1. [13] A. Dalgarno. C. Bottcher and G. A. Victor. Chem. 151W.A;,Bipgel.‘$I;
Phys.
Letters
5 (1970) 265..
[14] R. L. Smith and R. W. LaBnhn.
to be published.
CHEMICAL PHYSICS LETTERS
1 December
:
-:.
<. .
197,
. ,,
‘. ,:.
,F’
CALCULATIONS
.PSEUD,OPOTENTTAL -
OF
ALKALI
INTERACTIONS
J. N. BARDSLEY * Physics Deparlment, University of Pillsbargh. Pittsburgh. Pennsylvania 15213. VSA
. . .._
Received
6 October
1970
Several forms of pscudopotentinl arc presented for the valence electron in lithium and sodium. tentials are used to cakulate potential energy curves for Liz and Lip. Comparison with ab initio confirms the usefulness of the pseudopotentiai technique.
Many properties
of alkali
ly on the behaviour
atoms
of the valence
depend
main-
electron_
How-
ever, for ab initio calculations of the motion of the valence electron one needs to know the wavefunctions of all the core electrons. This is because the nuclear charge is partially screened by the surrounding core electrons, and because the valence electron wavefunction must be orthogonal to the core functions. This requirement of orthogonality causes considerable complication in the calculation of inter-atomic potentials. Formally the orthogonality requirement can be avoided by the introduction of a repulsive pseudopotential which is added to the potential experienced by the valence electron [ 11. The use of pseudopotentials is now a standard technique in solid state theory [2] and is becoming popular in atomic and mol_ecular physics. The ab initio calculation of pseudopotentials has been pioneered by Szasz and McGinn [3]. An alternative approach is to construct the pseudopotential empirically (e.g. as in refs. [4] and [5]). Further references and a description of the techniques can be found in the review by Weeks et al. [6]. In this
paper
the
empirical
approach
is illus-
trated and some preliminary results presented on the potential curves for Lii and Li2. The effective potential experienced by the valence electron in an alkali is expressed in the form 1 ‘Yd Veff(r) = -7 2
% 2(r2 + d2)3 where
.
c!d and try9 are
the
dipole
+ VSR(r) , and quadrupoie
(1)
These pocalculations
polarizabilities of the core and d is a cut-off parameter which is roughly equal to the radiuz of the core. VsR( r) is a short range potential which is small outside the core of the alkali; it is given a simple form with parameters which are chosen so that the eigenvalues of the one Vef (r) electron hamiltonian with the potential reproduce closely the observed alkali spec i rum. A significant difference between Ve;erf(r) and the Hartree-Fock potential for the aikali is that the former has no eigenvalues corresponding to the core orbitals. Thus for lithium the lowest s-wave eigenvalue of Veff( r) is equal to the en&gy of the valence electron in the 2s state. The purpose of the effective potential is both to represent the partially screened attraction of th nucleus and to simulate the influence of the Pauli principle on the valence electron. Becaus’ of the latter role Veff(r) depends strongly on th angular momentum of the valence electron. For example in lithium an s-wave electron is effectively repelled from the nucleus because of the requirement of orthogonality to the 1s function of the core, whereas for any non-zero angular momentum the Pauli principle leads to no constraint on the wavefunction. Thus we write
In general
for the partial wave I, the potential if there are core electrons wit 2H angular momentum I, and is attractive otherwise. The effective potential defined by eq. (1) is
Vz (r) is repulsive
* Most of this work wss cnrried
out :tt the University of TCXIS at Austin. The author is grntcful to that univcrsitv for the provision of free computer time. 51
Volume 7. number
CHEMICAL PHYSICS LETTERS
5
C+tlatio_n$ of the form
correct asymptotically to order l/r6 if ihe core polarizabilities are known and if d is chosen correctly. Thus if we integrate the one particle Schrodinger equation with this potential inwards from 00using the correct energy, the wavefunction will have the proper form outside the core. The normalization may be incorrect. However, there is a theorem in Quantum Defect Theory [7] that relates the normalization of a valence electron wavefimction to the rate of change of quantum defect with energy as one moves through a Rydberg series. Thus if one ensures that the effective potential leads to good energies for all the members of each Rydberg series then one obtains accurate approximations to the valence wavefunctions outside the core. This can be accomplished by introducing two (or more) variable parameters into each radial potential v;R(1-).
:
State BS
Gs
Ir~cl
fitting
for sodium.
GP GP
_
5rI
Gd
=-Ar&xp,kS~
VP) .
1
series the ?a=3nnd n=4 levels
In each
Effective principzt Esponcnt in1 -_--_. --. --3.64743 4.64930
Spectroscopic _______________._-..-._---3.64735 4.tiQS18
y&(r)
1970
(2) Several choices of_integers have been tried forb and q; a simple exponential (p = 0; q = l), a Hellmann or ,Yukawa potential (#= -1; q = l), and a modified gaussian form (p = 2; q = 2). The fitting of the exponential energy levels is illustrated in table 1 which applies to sodium. In each case the parameters Al and 51 are chosen so th’_t the lowest two levels of each series are reproduced exactly (spin-orbit effects are ingnored). The effective principle quantum numbers of the next two levels of each series are given in table 1. The experimental spectrum. can be reproduced with high accuracy (0.1% in ionization or excitation energies) with each form and thus the choice of the type of potential should be made on other grounds. A check of the empirical approach
Table
Spcctml
.’ :I December
1’ have been m,adeIvith potenti,als i
hsve
been
fitted
csnctl!
quantum numhers Hcllmnnn
Modified gnussinn
3.64743 4.ci4S30
3.64739 4.64924 4.13837
..-
4.13838 5.1407’3
4.13869
4.13869
5.14111
5.14113
5.14077
4.98G79 5 . 9dG”S _
4.98680 5.98G30
4.98GSlJ 5.98630
4.S8681 5.98G31
_____-.__-__
_.____ ____.-.__._._ _______ ___________________._----_-. -___-------
Table 2 t~seudopotentinl pnrnmeters __ _.__ ----..-.__----
--_-_________ 1
Espnncnt
in1
gaussinn
Cl
AI
52
AI
Cl
56.5046
3.61625
26.7618
2.R9636
15.7672
2.04616
1
-5.lSIlZO
3.JtiSW
-1.854Gl
Z.(i756(i
-1.86840
2.1ti726
> 2
-1.00861
3.10241
2.G1247
-0.025134
0.86045
----
ad=0.1325
-.--.-
--
1
_-_-_-_--__--..-
aq=o.112 --------_--_
I%eudtrpotcntinl
d=O.75 --_
Tnblc 3 pnrnmetcrs for sodium -._ _-________
4.53358
Al ---.--__-___
___-_--A-------
lti4.89G
Modified Cl
AI
3.85772
28.8425
39.7714
2.92164
2G.3684
i.40297
-3.33147
2.19462
-2.78823
1.79360
cYd=o.945
-_-----
ticlimnnn (1 ---_
--.--.-.-_-------0 317.512 1
O.G9607
Esponentinl AI
22
518
Modified
0
---_
~.----_
-.
_____-_----____~-
Hcllmann
Al
___
for lithium
act = 1.623 _-_-___--__
gsussinn _-_-
51 2.16472
3.00436
1XJDOl
--0.109$9
0.55401
d=l.l --+-
______------
--
Vo[rime ii. number 5
CHEMICAL
PHYSICS LETTERS
that the result of any calculation should not depend stror&y on the form of pseudopotential used. The potential parameters for Li and Na are given in tables 2 and 3. .‘ For‘calcdlating the interaction of an alkali ion with a neutral alkali we consider the system ABf and construct effective potentials Vsff( r) and Vzff (r). The effective hamiltonian is then
1 Dcccmbcr
ii
II= -+v;+
VA eff( la ) + r’gff( b) ‘
+ Vc (R) v
(3)
represents the interaction bein which V,(R) tween the two cores separated by the vector R; ra and rh are the vectors between the valence electron and the two nuclei. In the calculations reported here Vc(R) has been taken to he l/R, neglecting core-core polarization and overlnpping of the two cores. At each separation R the electronic energy is calculated by the variational method with a trial wavefunction of the form *(I-) = c 4,1 &#-a) nz The basic atomic of the equation {-sv;
functions
+ c b,, +.yb) n ti$l(?;)
(4)
are solutions
+ hp$ff(*-a) - CA} 6$,(ra)
=9
(5)
and the @E[(rb) are defined in an analogous manner. The energy CA is taken to be the ground state energy of A. The scaling parameter X is treated as an eigenvalue for the generation of the basic functions. Atomic functions defined in this way form a complete set without any continuum. This method has been applied t&the calculation of the lowest 2Zi and 2s; stat& of Lii . The results using six expansion functions for each atom are given in table 4. The dissociation energy of 1.23 eV and the equilibrium nuclear separation of 5.9 au are close to the predictions of ab initio calculations [8,91. The technique can easily be extended to systems with several valence electrons. The results for the lowest I>* and 3X, states of Liz are also shown in tableif: The accuracy of the calculation is limited by the number of two electron-integrals that can be computed at a reasonabIe cost. Using a 13 configuration wavefunction the dissociation energy is calculated to be 0.89 eV, whereas the experimental energy is 1.05 eV [lo] and an ab initio calculation gives 0.99 eV [ 111. The equilibrium separation is close to the experimental value of 5.05 au. Before a final
Vdencc
R
Tal~lc 4 cncrgics for 12, :fnd Lip. tics nrc in :llomic units
electron
Li 2?x,
1970
All qwrnti-
Liz 2,.
R
-‘II
1x
R
3s .‘,I
6
-0.23SB
-0.1105
6.5
-0.2429
-0.1246
5.5
-0.4278
G
-0.2435
-0.1366
ti
-0.4241
-0.3847
6.5
-0.2421
-O.146!1
8
-~J.JlJ(il
-0.3!J.U
-0.2305
-0.1557
10
-0.3990
-0_3!Ki5
-0.23(i3
-0.1633
12
-0.3973
-0.3969
-I).1 OY2
-ll.l9S2
00
JJ.39ti3
-u.39ti3
7 ‘i..i
00
5
-0A200
-0.3790
judgement can be made on the technique the numerical techniques must be improved so that more configurations can be included. One problem that should be investigated by this method is the range of applicability of asymptotic forms of the long range forces between atoms. The calculations on Li2 show that the effects of overlap of electron orbit&s are important out to separations of 15 au which is three times the equilibrium separation_ Kutzelnigg [51 has shown that it is at similar separations that the Rs8 term in the multipole expansion becomes small in comparison with the pure van deriSVaals attraction, These facts may explain th’Z large difference between calculated van der Waals coefficients for alkalis and those
derived from scattering data [ 121. A different use of empirical potentials
for alkali valence electrons has been reported by Dalgarno et al. [13]. Their empirical potential does not simulate the effect of the Pauli principll momentum. and thus is not dependent on an@ar The eigenfunctions of this potential approximate both core and valence orbitnls, and the problem of orthogonality of core and valence functions remains. An advantage of their approach is that the valence functions have the correct number of nodes and thus may be better approximations inside the core region. They also treat the corecore interaction more carefully than has been
done here.
In conclusion it is our belief that empirical pseudopotential method can be used to obtain accurate potential curves but further work is needed to demonstrate this thesis. In choosing an empirical potential an explicit f-dependence should be introduced and at least two adjustable parameters included in each component. There are also many possible applications of the meth51!
Volume
7. .number
CHEMICALPHYSIC!S.~TTE~
5
__-. ‘:
_..
od to electron
and
photon
collisions
with
atoms;
as shown by the photoionization calculations of Smith and LaE!ahn [14].
REFERENCES [lj J. G. Phillips and L. Kleinman. Phys. Rev. 116 (19.59) 287; 118 (1960) 1351. I21 Pscudcmotcntinls in the theory of . W. A. Harrison. metals (Benjamin. Now-York. 1966). [31 I+Szasz and 0. McGinn. J. Chem. Phys. 45,(1966) 2898: 47 (1967) 3495: 43 (1969) 2997: L. R. Kahn and W. A.Goddard Ill. Chem. Phgs. Letters 2 (1968) 667. [4J L.Sznsz and G. BIcGinn. J. Chem. Phys. 42 (1965) 2363; J. Csllnwvny and P.S. Lnghos. Phys. Rev. 187 (1969) 192: G.Simons and A. Mazziotti. J. Chem. Phys. 52 (1970) 2-149. I
520
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Phys.
Letters
5 (1970) 265..
[14] R. L. Smith and R. W. LaBnhn.
to be published.