INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. IX, 215-227 (1975)

A First-Principle Pseudopotential Calculation of the (ns) Energy Levels of the K Atom. 11.

P. CSAVINSZKY AND R. HUCEK Department o f Physics, University o f Maine, Orono, Maine 04473, U.S.A.

Abstracts

We have calculated the energies of the (7s) and (8s) states of the valence electron in the K atom by making use of a pseudopotential based on recently obtained variational solu- tions of the Thomas-Fermi (TF) equation for neutral atoms, and positive ions. We have chosen trial wave functions, with appropriate parameters, for the valence electron and then minimized the energies of the respective states using the pseudoHamiltonian. The exchange interaction between the K+ core and the valence electron has also been considered as a perturbation. Comparison of the calculated (7s) and (8s) energies with the experimental values shows an agreement of about 6% for the former one and of about 5% for the latter one, respectively. With the exchange correction both of these discrepancies are reduced to less than 2%. We conclude that the procedure outlined here is a promising one in dealing with problems involving a highly excited electron outside of a closed-shell ion core, a system for which a more exact quantum-mechanical treatment would be much more difficult.

Nous avons calcult les tnergies des ttats 7 s et 8s de l’tlectron de valence de l’atome de K en employant un pseudopotentiel bast sur des solutions variationnelles de l’tquation de Thomas-Fermi pour des atomes neutres et des ions positifs, obtenues rkcemment. Nous avons choisi des fonctions d’onde d’essai avec des param6tres convenables pour minimiser les Cnergies des 6tat.s en question A l’aide du pseudo-Hamiltonien. L’interaction d’kchange entre le coeur K+ et l’tlectron de valence a t t t considtrtee comme une perturbation. La comparaison des tnergies calcultes et exptrimentales montre un accord d’environ 6% pour 7s et 5% pour 8s. Avec la correction d’kchange les deux differences sont reduites a moins de 2 % . Nous en concluons que le procedt employe ici donne des esperances pour le traitement d’un electron hautement excite au dehors d’un coeur d’ion a couches complttes, systkme pour lequel un calcul plus exact serait beaucoup plus difficile.

Wir haben die Energien der 7s-und 8s-Zustande des Valenzelektrons im K-Atom mittels eines auf neulich erhaltene Variationslosungen der Thomas-Fermi-Gleichung fur neutrale Atome und positive Ionen basierten Pseudopotentials berechnet. Versuchswel- Ienfunktionen mit angemessenen Parametern wurden gewahlt, und die Energien der jeweiligen Zustande wurden mit Hilfe des Pseudo-Hamiltonoperators minimisiert. Die Austauschwechselwirkung zwischen dem Kf-Rumpf und dem Valenzelektron ist als Storung betrachtet worden. Der Vergleich zwischen den berechneten und den experi- mentellen Energien zeigt eine Ubereinstimmung von ungefahr 6% fur 7s und 5% fur 8s. Mit der Austauschkorrektur werden beide Unterschiede zu weniger als 2% reduziert. Das hier angegebene Verfahren scheint also versprechend zu sein fur Probleme mit einem hoch angeregten Elektron ausserhalb einem Ionenrumpf, rnit abgeschlossenen Schalen, einem System fur welches eine exaktere quantenmechanische Behandlung vie1 schwieriger ware.

215

0 1975 by John Wiley & Sons, Inc.

2 16 CSAVINSZKY AND HUCEK

1. Introduction

I n a previous paper [l] (hereafter referred to as I) we have calculated the energies of the (4s), (5s), and (6s) states of the valence electron in the K atom by making use of a pseudopotential based on recently obtained variational solutions of the Thomas-Fermi (TF) equation for neutral atoms, and positive ions. The present work reports on an extension of these calculations to the highly excited (7s) and (8s) states. As in I, trial wave functions with two parameters have been chosen for the valence electron and then the energies of the respective states have been determined by finding the energy minima with respect to these parameters. As in I, the exchange interaction between the K+ core and the valence electron has also been considered as a perturbation. Comparison of the variational (7s) and (85) excited states energies with the experimental values shows an agreement of about 6 and 5 %, respectively, which, with the exchange correction, is reduced to less than 2 %. As in I, we conclude that the procedure outlined here is a prom- ising one in dealing with problems involving a highly excited electron outside of a closed-shell ion core, a system for which a more exact quantum mechanical treatment would be much more difficult.

2. Calculation

We have obtained variational solutions of the one-electron Schrodinger equation (in atomic units)

(-$ V2 - eoVmod)y(ns) = E(ns)y(ns)

where the modified potential (i.e., the pseudopotential) is of the form

(2) Vmo, = vi + F n s

with Vi standing for the potential of the K+ ion, and FnS for a nonclassical poten- tial that replaces the Pauli principle. Detailed analytical expressions for the quan- tities appearing in Equation (2) are given in Paper I.

We have chosen the trial wave functions (in atomic units) as

(3)

and

(4) ~ ( 8 s ) = A(8s)r'(8s)e-1(8s)r(1 + Cgr + C i r 2 + CI;'r3 + Cry4)

where k and 1 are variational parameters, the A's are normalization constants, and the C's are constants determined from the requirement that the wave function of a given s-state of the valence electron should be orthogonal to all the other lower-lying s-states. Detailed expressions for these quantities are given in the Appendix [2].

y(7s) = A(7s)r"7"e-"7s'"(1 + C;r + C;'r2 + Cyr3)

PSEUDOPOTENTIAL CALCULATION OF ?tJ ENERGY LEVELS OF K ATOM 217

( 5 )

and

( 6 )

Solutions to Equation (1) have been obtained by imposing the conditions

-- - 0 83. ( ns)

where the subscript u on E stands for the word variational. The results of the calculations are summarized in Tables I and 11, the former

TABLE Ia. Parameter values used in the calculations.

k 3, R R' (477A2)-' State ( a 3 (UH) (UH) ( 4 + 3 )

( 7s) 1.52 0.218 1.890 6.604 1530 (85) 1.46 0.179 1.890 6.609 2382

TABLE Ib. Parameter values used in the calculations.

G c; c"; State (62 ( a 3 (a??,

-0.2671 0.01980 - 0.0004 188 (75)

TABLE Ic. Parameter values used in the calculations.

G State

-0.1985 0.02774 -0.0009774 0.00001 126 (8s)

TABLE 11. Energy values obtained by using the pseudopotential in Equation (2).

State

(73) -0.5870 -0.5504 6.24 (85) -0.4022 -0.381 2 5.22

containing the parameters, and the latter the energy values. The quantities displayed in Table I need little comment, with the exception

of R and R'. As discussed in I, the ion potential, Vi , at large distances from the

2 18 CSAVINSZKY AND HUCEK

nucleus, does not show the correct asymptotic behavior. I t falls off as (Ze,/r)qi (where qi is the solution of the TF equation for the K+ ion), whereas it should fall off as ( e J r ) . For this reason, we have introduced a cut-off distance R determined from

(7) Z e 0 e0 - qz(r ) = - r r

so that

As in I, we have followed Comb& who considered the exchange interaction between the ion core and the valence electron as a perturbation. For this inter- action he has derived the expression

(9) n

where y is a constant defined in I, and the quantities p E and pns stand for the elec- tron density of the ion core, and the electron density of the valence electron, respectively.

Considering that in the ion core pns << p i , and that far from the nucleus pi << pns , we can determine a distance R’(ns) from

(10) pi ( ’ ) = Pns(r )

and expand the quantity (p i + pnJ4i3 into a binomial series for both of the limiting cases. Breaking off the expansions after the second term, we obtain for EA in Equation (9)

(11)

where the detailed forms for the latter quantities are given in I.

are listed in Table 111.

E, = E:’ + E:’

The energies of the (7s) and (8s) states, modified with the exchange interaction,

TABLE 111. Energy values corrected by the exchange interaction between the K+ core and the valence electron.

(75) -0.04572 -0.00002386 -0.04574 -0.5961 1.55 (85) -0.02813 -0.00002007 -0.02815 -0.4094 1.79

PSEUDOPOTENTIAL CALCULATION OF ns ENERGY LEVELS OF K ATOM 219

Finally, it is of some interest to see how large is the radius of the pseudo- orbit. We have calculated this quantity from

and listed the values in Table IV. This table contains entries for the (4s), (5s),

TABLE IV. Expectation values of the radii of the pseudoorbits.

( r ) State (UH)

5.955 13.167 23.528 37.026 53.399

and (6s) states too, since (r),,s has not been given in I. The analytical expressions for ( r )ns are given in the Appendix.

3. Discussion

Inspection of the data in Table I1 reveals that the discrepancy between the experimental and variational energies of the (ns) states follows a diminishing trend as n increases. The same conclusion has also been drawn in I for the lower- lying states. This finding is expected since approximations made in describing the core, such as the use of the TF model which neglects the exchange effect, should appear less severe for higher excited states of the valence electron. For such states the core appears more and more as a point charge so that the finer details of the electron distribution within the core become less important. Inspection of the data also shows that the calculated energy values lie above the experimental ones, as they must in a variational calculation.

Looking at the energy values in Table 1x1, we see that the inclusion of the exchange interaction between the K+ core and the valence electron leads to a significant reduction of the discrepancy between the experimental and the cal- culated energy values. The data also reveals that E, + E, is now slightly below the E,,, values. This, however, is permissible since E, is a perturbation.

The question may be asked of just how good an approximation is [E:) + E:’] to EA . To answer this question we have evaluated the exact ex- pression for EA , Equation (S), by the technique of Gaussian quadratures. The exact values of EA , together with the deviation of [Ez’ + EZ’] from E, , are listed in Table V for the (4s) to (8s) states. (These data were not given in I for the (4s) to (6s) states.) I t is seen from Table V that [E:’ + Ey’] is a good ap- proximation to EA (exact), for all states considered.

220 CSAVINSZKY AND HUCEK

TABLE V. Comparison of the exact values of the exchange correction with the approximate ones.

EA(exact) State (eV)

(45) -0.4173 -0.3913 (55) -0.1820 -0.1741 (6s) -0.08599 -0.08379 ( 75) -0.04627 -0.04574 (8s) -0.02830 -0.02815

-0.0260 -0.0079 -0.00220 -0.00053 -0.00015

Throughout the calculation of the (ns) states of the valence electron we have assumed that the atom is composed of an ion core interacting with the valence electron. This approximation should be good if there is a negligible probability of finding the valence electron within a radius R, the distance at which the ion potential becomes that of a singly charged ion, namely eo/r. To check on the validity of this assumption, we have calculated 477 jF p,,r2 dr. The values ob- tained for the (4s) to (8s) states are displayed in Table VI. (These values have not been given in I for the (4s) to (6s) states.) Inspection of the data in Table VI shows that, even for the ground state associated with the orbit

TABLE VI. Probability of finding the valence electron within a sphere of radius R.

State

0.8185 0.3748 0.1795 0.09823 0.06107

of least radius, the electron density of the valence electron within a sphere of radius R is negligible. For this reason, the model of the atom consisting of an undistorted ion around which there is a valence electron appears to be a good one.

Appendix Constants Occurring in the Wave Functions

This lengthy Appendix has been included because the expressions derived for the (7s) and (8s) states of the valence electron in the K atom have a universal applicability in the sense that they are also valid for the third and fourth excited states of any atom or ion that has an s-electron in these states outside of a closed- shell ion core. (Examples are the (6s) and (7s) states of the valence electron in

PSEUDOPOTENTIAL CALCULATION OF T2.Y ENERGY LEVELS OF K ATOM 221

(-4.3) D,C; =

D,C‘; =

D,CY =

the Na atom, or the (8s) and (9s) states of the valence electron in the R b atom, etc.) For exactly the same reason as that outlined above the formulae in the Ap- pendix of paper I have also universal applicability.

The normalization constants are given by

L4 L2 L3

L8 L, L,

Ll2 Ll, Lll

Ll L4 L3

L9 Ll, Lll

Ll L2 L4

L9 4 0 LIZ

L, L8 L,

L5 L, L8

222 CSAVINSZKY AND HUCEK

(A.6) D7 =

Ll L2 L3

L9 Ll, Lll

L, L, L7

x7

u7 (A.7) L, = -

(A.lO) L, = -1

PSEUDOPOTENTIAL CALCULATION OF ns ENERGY LEVELS OF K ATOM 223

where, in Equations (A.7) to (A.17), the following additional definitions have been used

(A. 18)

(A. 19)

(A.20)

(A.22)

X7 = k7 + k4 + 3 Y7 = k7 + k5 + 3 Z7 = k7 + ks + 3

v7 = 1 7 + 1 5

(A.21) u7 = 1, + 1,

(A.23) w7 = 1 7 + 1 6

Similarly, the orthogonality constants in g(8s), resulting from the orthogonality requirements, are given by

M5 M2 M3 M4

Mio M7 Ma M9

M20 MI7 MI8 Ml9

MI M5 M3 M4

Mi, Ma M9

(A.24) D,CL = M15 M I Z M13 M14

(A.25) DaC: = M15 M13 M14

M16 M20 M18 M19

MI M2 M5 M4

M6 M7 M9 (A.26) D,C;;' =

M12 M15 M14

M16 M17 M20 M19

MI M2 M3 M5

M7 Ma Mio

MI, M12 4 3 Ml5

D C = (A.27) 8 a

M16 M17 M20

where, in Equations (A.24) to (A.27), D, is defined by

JMl M2 M3 M4 I (A.28)

224 CSAVINSZKY AND HUCEK

The Mi's appearing in Equations (A.24) to (A.28), are defined by

PSEUDOPOTENTIAL CALCULATION OF n.r ENERGY LEVELS OF K ATOM 225

(A.42)

(A.44)

(A.45)

(A.46)

(A.47)

Ml8 = +

226 CSAVINSZKY A N D HUCEK

where, in Equations (A.29) to (A.48), the following additional definitions have been used:

(A.49)

(A.50)

(A.51)

(A.52)

(A.53)

x8 = k8 + k5 + 3

ys = k8 + k4j + 3

2 8 = k, + k7 + 3

w8 = k8 + k4 f 3

T8 = 1 8 + 1, (A.54)

(A.55)

(A.56)

US = 1 8 + 1 6

v 8 = a8 + 1 7

s, = ?L8 + a, The expectation values for the radii of the (4s) to (8s) states are given by

(A.57)

(A.59)

(A.60)

PSEUDOPOTENTIAL CALCULATION OF ?ZJ ENERGY LEVELS OF K ATOM 227

(A.61)

Bibliography

[ I ] 1'. Csavinszky and R. Hucek, Int. J. Quantum Chem.: Symposium No. 8,37(1974). (The work pertaining to the (4s) to (6s) states was presented as a short invited paper at the International Symposium on Atomic, Molecular, and Solid State Theory, and Quantum Statistics, Sanibel Island, Florida, January 20-26, 1974. The work pertaining to the (7s) and (8s) states was presented as a contributed paper at the meeting of the American Physical Society, Washington D.C., April 22--25, 1974, and published in Bull. Am. Phys. Soc. 19, 571 (1974).) To keep Paper I1 as brief as possible, the reader should consult Paper I for all details and references.

Received May 14, 1974. Revised September 5, 1974.