Core Ionization Energies of Atoms andIons Calculated Using the GeneralizedSturmian Method

JOHN AVERY, RUNE SHIMH. C. Ørsted Institute, University of Copenhagen, Copenhagen, Denmark

Received 27 December 1999; accepted 13 March 2000

ABSTRACT: The generalized Sturmian method for solving the many-electronSchrödinger equation is reviewed. The method is illustrated with calculations of the coreionization energies of a series of atoms and ions. It is shown that when the “basispotential” is chosen to be the actual attractive potential of the nuclei in the system beingstudied, convergence is rapid, and a correlated solution can be obtained without the use ofthe self-consistent field approximation. Furthermore, when many-electron basis functionsof this type are used, the kinetic energy term disappears from the secular equation, thenuclear attraction potential is diagonal, and the Slater exponents of the basis functions areautomatically optimized. c© 2000 John Wiley & Sons, Inc. Int J Quantum Chem 79: 1–7, 2000

Key words: atomic physics; quantum theory; core ionization energies; Sturmians;excited states

Generalized Sturmian Basis Sets

T he generalized Sturmian method, which hasbeen described in our previous publications

[1 – 8], is a form of direct configuration interaction,with a special prescription for preparing optimalconfigurations. It can be applied both to atoms andto molecules, but in the present article we shall con-fine the discussion to atoms and atomic ions. Themethod makes use of a basis set in which each basisfunction is an N-electron configuration, φν(x), espe-cially prepared so that it is a solution of the equation

Correspondence to: J. Avery.

(in atomic units):[−1

2

N∑j= 1

∇2j + βνV0(x)− E

]φν(x) = 0. (1)

In Eq. (1), N is the number of electrons in the system,∇2

j is the Laplacian operator of the jth electron, andV0(x) is the attractive Coulomb potential of the nu-clei in the system. In the present article we consideratoms and atomic ions, so that

V0(x) = −N∑

j= 1

Zrj

. (2)

(In the case of molecules, V0(x) would be the many-center attractive Coulomb potential produced by allthe nuclei in the molecule.) The constants βν are

International Journal of Quantum Chemistry, Vol. 79, 1–7 (2000)c© 2000 John Wiley & Sons, Inc.

AVERY AND SHIM

chosen in such a way that all of the configurationsin the basis set correspond to the same value of E.In other words, each configuration can be thoughtof as a solution of the many-electron Schrödingerequation for a set of N noninteracting electrons inthe attractive potential of a nucleus with effectivecharge Qν = βνZ, the effective charges being chosenin such a way that all of the functions in the basis setcorrespond to the same value of the energy. Such aset of functions is easy to construct if we make useof the familiar hydrogenlike spin orbitals,

χµ(xj) = Rnl(rj)Ylm(θj,φj){α(j)β(j) (3)

where µ stands for the quantum numbers {n, l, m, s}and

R1,0(rj) = 2Q3/2ν e−Qν rj ,

R2,0(rj) = 1√2

Q3/2ν e−Qν rj/2

(1− Qνrj

2

),

R2,1(rj) = 1

2√

6Q5/2ν e−Qν rj/2rj

... (4)

The functions χµ(xj) are solutions to the one-electronSchrödinger equation for an electron in the attrac-tive potential of a nucleus with charge Qν :[

−12∇2

j −Qν

rj+ 1

2

(Qν

n

)2 ]χµ(xj) = 0. (5)

Now suppose that we let

φν(x) ≡ |χµχµ′χµ′′ · · · |

≡ 1N!

∣∣∣∣∣∣∣∣χµ(x1) χµ′(x1) χµ′′ (x1) . . .

χµ(x2) χµ′(x2) χµ′′ (x2) . . ....

......

...χµ(xN) χµ′(xN) χµ′′ (xN) . . .

∣∣∣∣∣∣∣∣ . (6)

If we act on this function with the kinetic energyoperator of the N-electron system, then Eq. (5) tellsus that the result will be[−

N∑j= 1

12∇2

j

]φν(x) =

[Qν

r1− 1

2

(Qν

n

)2

+ Qν

r2

− 12

(Qν

n′

)2

+ · · ·]φν(x)

=[−βνV0(x)− Q2

ν

2

(1n2 +

1n′2

+ 1n′′2+ · · ·

)]φν(x). (7)

(The Slater determinant is a sum of terms, related toeach other by permutations of the electron indices,

but the N-electron kinetic energy operator acting oneach of these terms brings out the same factor, re-gardless of the permutations.) Looking at Eq. (7), wecan see that if we wish φν(x) to satisfy Eq. (1), thenwe must choose Qν (and hence βν = Qν/Z) in sucha way that

E = −Q2ν

2

(1n2 +

1n′2+ 1

n′′2+ · · ·

). (8)

Potential-WeightedOrthonormality Relations

If we multiply Eq. (1) from the left by the con-jugate of another configuration from the basis set,φν′ (x), and integrate over the coordinates of the N-electron system, we obtain∫

dx φ∗ν′(x)

[−1

2

N∑j= 1

∇2j +βνV0(x)−E

]φν(x) = 0. (9)

Similarly, interchanging the indices ν and ν ′, wehave∫

dx φ∗ν (x)

[−1

2

N∑j= 1

∇2j + βν′V0(x)− E

]φν′ (x) = 0.

(10)We next take the complex conjugate of Eq. (10) andsubtract it from (9), making use of the hermiticity ofthe matrix∫

dx φ∗ν′(x)

[−1

2

N∑j= 1

∇2j − E

]φν(x). (11)

This gives us the relationship

(βν − βν′ )∫

dx φ∗ν′(x)V0(x)φν(x) = 0 (12)

from which it follows that∫dx φ∗ν′ (x)V0(x)φν(x) = 0 if βν 6= βν′ . (13)

To normalize the configurations in our basis set,we recall that the familiar hydrogenlike orbitals ofEqs. (4) obey the relationship∫

d3xj∣∣χµ(xj)

∣∣2 1rj= Qν

n2 . (14)

Then, making use of the Slater–Condon rules andEq. (8), we obtain∫

dx∣∣φν(x)

∣∣2V0(x) = −Qν

βν

∑µ⊂ν

∫d3xj

∣∣χµ(xj)∣∣2 1

rj

2 VOL. 79, NO. 1

CORE IONIZATION ENERGIES

= −Q2ν

βν

(1n2 +

1n′2+ 1

n′′2+ · · ·

)= 2Eβν

. (15)

Thus the potential-weighted orthonormality rela-tions obeyed by the configurations in our many-electron Sturmian basis set become∫

dx φ∗ν′(x)V0(x)φν(x) = δν′ ,ν 2Eβν

. (16)

The Generalized SturmianSecular Equation

Having constructed our generalized Sturmianbasis set, each basis function of which is an N-electron configuration, we would now like to use itto solve the Schrödinger equation,[

−12

N∑j= 1

∇2j + V(x)− E

]ψ(x) = 0, (17)

where

V(x) = V0(x)+ V′(x) (18)

and

V′(x) =N∑

i>j

N∑j= 1

1rij

. (19)

If we represent the N-electron wave function, ψ(x),by a linear superposition of configurations, then (17)becomes∑

ν

[−1

2

N∑j= 1

∇2j + V(x)− E

]φν(x)Bν = 0. (20)

Since each configuration satisfies Eq. (1), and sinceall of the configurations correspond to the samevalue of the energy, E, Eq. (20) can be rewritten inthe form ∑

ν

[V(x)− βνV0(x)

]φν(x)Bν = 0. (21)

Multiplying from the left by a conjugate functionfrom the basis set, integrating over the coordinatesof the electrons, and making use of the potential-weighted orthonormality relations, (16), we obtain∑

ν

[∫dx φ∗ν′ (x)V(x)φν(x)− 2Eδν′ ,ν

]Bν = 0. (22)

It is now convenient to introduce the definitions,

p20 ≡ −2E (23)

and

Tν′ ,ν ≡ − 1p0

∫dx φ∗ν′ (x)V(x)φν(x). (24)

It turns out that when V(x) is given by (18) and (19)and when the configurations φν(x) are constructedin the way which we have described, then thematrix Tν′ ,ν defined by (24) is independent of p0.Using (23) and (24), we can rewrite the secular equa-tion, (22), in the form∑

ν

[Tν′ ,ν − p0δν′ ,ν]Bν = 0. (25)

If we introduce the further definitions,

Rν ≡(

1n2 +

1n′2+ 1

n′′2+ · · ·

)1/2

(26)

and

T0ν′ ,ν ≡ −

1p0

∫dx φ∗ν′ (x)V0(x)φν(x), (27)

then (23) and (8) can be combined to yield

p0 = QνRν = βνZRν (28)

and from the potential-weighted orthonormality re-lation, (16), we obtain

T0ν′ ,ν = δν′ ,νZRν . (29)

Thus, finally, the generalized Sturmian secularequation takes on the form,∑

ν

[T′ν′ ,ν − δν′ ,ν(p0 − ZRν )

]Bν = 0, (30)

where

T′ν′ ,ν ≡ −1p0

∫dx φ∗ν′ (x)

N∑i>j

N∑j= 1

1rijφν(x). (31)

In the crude approximation where a state of theN-electron system is represented by a single config-uration (i.e., a single basis function), the generalizedSturmian secular equation reduces to the require-ment that

p0 ≈ ZRν − |T′ν,ν | (32)

(T′ν,ν being negative). From Eq. (28), it then followsthat the effective nuclear charge characterizing theconfiguration will be approximately

Qν ≈ Z− |T′ν,ν |Rν

. (33)

Equation (23) tells us that the energy of the configu-ration will be roughly

Eν ≈ −12

(ZRν − |T′ν,ν |

)2. (34)

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 3

AVERY AND SHIM

The approximation shown in Eq. (34) improves as Zbecomes large, while for Z < N it deteriorates andshould not be used.

Core Ionization Energies

To illustrate the general principles discussedabove, we have calculated the core ionization en-ergies of a number of atoms and ions in the crudeone-basis-function approximation. From Eq. (34),we can see that in this rough approximation, theenergy of the ground state or an excited state ofan atom or ion can be expressed as a quadraticfunction of the nuclear charge, Z. For example, theground-state energies of the first few N-electron iso-electronic series are given approximately by

E0 ≈ − 12

(Z√

21 − 0.441942

)2, N = 2

E0 ≈ − 12

(Z√

21 + 1

4 − 0.681870)2

, N = 3

E0 ≈ − 12

(Z√

21 + 2

4 − 0.993588)2

, N = 4

E0 ≈ − 12

(Z√

21 + 3

4 − 1.40773)2

, N = 5

E0 ≈ − 12

(Z√

21 + 4

4 − 1.88329)2

, N = 6

E0 ≈ − 12

(Z√

21 + 5

4 − 2.41491)2

, N = 7. (35)

This is a rough approximation, but it yields ener-gies in reasonably good agreement with Clementi’sHartree–Fock results [9], as shown in our previouspublications and illustrated in Figure 1. We now re-move an electron from the 1s orbital of the core of

FIGURE 1. This figure shows the ground stateenergies of the 6-electron isoelectronic series, C, N+,O2+, etc. The energies, expressed in Hartrees, areshown as functions of the nuclear charge, Z. The smoothcurve, calculated from Eqs. (35), can be compared withClementi’s Hartree–Fock results [9], which are shownas dots.

each atom or ion, so that (for example)

|χ2sχ1sχ1s| → |χ2sχ1s|, N = 3. (36)

We thus obtain a set of core-excited states for whichwe can calculate the approximate energy in thesingle-configuration approximation, using Eq. (34).This gives us the energies

Ei ≈ − 12

(Z√

11

)2, N′ = 1

Ei ≈ − 12

(Z√

11 + 1

4 − 0.168089)2

, N′ = 2

Ei ≈ − 12

(Z√

11 + 2

4 − 0.447600)2

, N′ = 3

Ei ≈ − 12

(Z√

11 + 3

4 − 0.807971)2

, N′ = 4

Ei ≈ − 12

(Z√

11 + 4

4 − 1.23995)2

, N′ = 5

Ei ≈ − 12

(Z√

11 + 5

4 − 1.73489)2

, N′ = 6. (37)

In Eqs. (37), N′ = N − 1 represents the numberof electrons after core ionization. From among thepossible eigenfunctions of the total spin operator,the core-ionized configuration has been taken to bethat of maximum spin. By subtracting (35) from (37),we obtain the approximate core ionization energiesin Hartrees as quadratic functions of the nuclearcharge, Z:

1E ≈ 0.097656− 0.625000Z+ 0.5Z2,N = 2

1E ≈ 0.218346− 0.834877Z+ 0.5Z2,N = 3

1E ≈ 0.393436− 1.02280Z+ 0.5Z2,N = 4

1E ≈ 0.664437− 1.26560Z+ 0.5Z2,N = 5

1E ≈ 0.993840− 1.49133Z+ 0.5Z2,N = 6

1E ≈ 1.39664− 1.73413Z+ 0.5Z2, (38)N = 7.

The approximate ionization energies of Eqs. (38) areshown in Table I and Figures 2 and 3 as functionsof N and Z.

The 3-Electron Isoelectronic Series

Although the single-configuration approxima-tion gives reasonable results for very little effort, itis of course desirable to improve it by adding moreconfigurations. This introduces a complication,

4 VOL. 79, NO. 1

CORE IONIZATION ENERGIES

TABLE ICore ionization energies in Hartrees for a numberof atoms and ions, calculated in the crude1-configuration approximation.a

N = 2 N = 3 N = 4 N = 5 N = 6

Z = 2 0.85Z = 3 2.72 2.21Z = 4 5.60 4.88 4.30Z = 5 9.47 8.54 7.78 6.84Z = 6 14.35 13.21 12.26 11.07 9.95Z = 7 20.22 18.87 17.73 16.31 14.95Z = 8 27.10 25.54 24.21 22.54 20.94Z = 9 34.97 33.20 31.69 29.77 27.93Z = 10 43.85 41.87 40.17 38.01 35.92

a In this table, the core-ionized state is always chosen, out ofthe possible spin eigenfunctions, to be the one which has thehighest value of total spin.

since each configuration is characterized by itsown effective nuclear charge, Qν . In evaluatingoff-diagonal elements of the matrix T′

ν′ ,ν betweentwo configurations characterized by different valuesof Qν , we cannot assume radial orthonormality ofthe spin orbitals, although spin orthonormalityand angular orthonormality remain unaltered. Ingeneral, evaluation of these off-diagonal matrixelements requires the use of the generalized Slater–Condon rules, as discussed in Refs. [10 – 14]. How-ever, the 3-electron series of atoms and ions presentsan especially simple case: If we make use of spinorthonormality, we can expand the Slater determi-nants directly without obtaining very many terms.For example, we can consider the configurations

FIGURE 2. This figure shows the approximate coreionization energies in Hartrees for various atoms andions. The 1-configuration approximation, Eqs. (38), wasused in calculating1E, which is shown as a functionof N (the number of electrons) for various values of Z.

FIGURE 3. This figure, like Figure 2, illustrates theresults shown in Eqs. (38). The approximatecore-ionization energies,1E, are shown as functionsof Z for various values of N.

φν(x) = |χaχbχc̄| =√

3!A[χa(1)χb(2)χc̄(3)

](39)

and

φν′ (x) = |χdχeχf̄ | =√

3!A[χd(1)χe(2)χf̄ (3)

], (40)

where A is the idempotent antisymmetrizingoperator

A = 1N!

∑P

εPP. (41)

In Eq. (40), χc̄ and χf̄ represent spin-down spinorbitals, while the remainder of the spin orbitalsin the two configurations are spin-up. Since theinterelectron repulsion potential V′(x) is totallysymmetric with respect to interchange of theelectron coordinates, it follows that∫

dx φ∗ν′ (x)V′(x)φν(x)

= 3!∫

dxA[χ∗a (1)χ∗b (2)χ∗c̄ (3)

]V′(x)

×A[χd(1)χe(2)χf̄ (3)]

= 3!∫

dxχ∗a (1)χ∗b (2)χ∗c̄ (3)V′(x)A[χd(1)χe(2)χf̄ (3)

]=∫

dxχ∗a (1)χ∗b (2)χ∗c̄ (3)[

1r12+ 1

r13+ 1

r23

]× [χd(1)χe(2)χf̄ (3)− χe(1)χd(2)χf̄ (3)

]. (42)

Only six terms survive when spin orthogonality istaken into account. Equation (42) can be rewrittenin the form,∫

dx φ∗ν′ (x)V′(x)φν(x)

= 〈c|f 〉〈a, b|g|d, e〉 − 〈c|f 〉〈a, b|g|e, d〉+ 〈b|e〉〈a, c|g|d, f 〉 − 〈b|d〉〈a, c|g|e, f 〉+ 〈a|d〉〈b, c|g|e, f 〉 − 〈a|e〉〈b, c|g|d, f 〉, (43)

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 5

AVERY AND SHIM

where 〈µ′|µ〉 is an overlap integral, and where

〈µ′′,µ′′′|g|µ,µ′〉 ≡∫

d3x1

∫d3x2 χ

∗µ′′ (x1)χ∗µ′′′ (x2)

× 1r12χµ(x1)χµ′(x2). (44)

We now let

φν(x) = |χnsχ1sχ1s| (45)

and

φν′ (x) = |χn′sχ1s′χ1s′ |, (46)

where

χns(xj) ≡ Rn,0(rj)Y0,0(θj,φj)αj. (47)

In Eq. (46), the prime on the 1s′ spin orbitalsserves to remind us that since the configurationφν′ (x) has an effective nuclear charge which differsfrom that belonging to the configuration φν(x), theSlater exponents of the 1s orbitals will differ inthe two configurations. Using (43) to evaluate theinterconfigurational matrix elements, we obtain

T′ν′ ,ν =

0.681870 0.040814 0.021497 . . .

0.040814 0.563128 0.028240 . . .

0.021497 0.028240 0.513826 . . ....

......

(48)

A 10×10 T′ν′ ,ν matrix was generated by letting n′ and

n take on the range 2, 3, . . . , 11, and Eq. (48) showsthe upper left-hand corner of this matrix. The matrixwas then substituted into the generalized Sturmiansecular equation, (30), and the equation was solved.This yielded the ground-state energies, E0, of the3-electron isoelectronic series, as shown in Table II.

In a similar way, we calculated Ei, the energy ofthe 3S core-ionized state obtained by removing anelectron from the core of the 3-electron isoelectronicseries, using 10 configurations of the form

φν(x) = |χnsχ1s| (49)

with n running from 2 up to 11. These energies arealso shown in Table II, and the difference,

1Ecalc ≡ (Ei)calc − (E0)calc, (50)

can be compared with experimental values takenfrom Moore’s tables [15]. The 10-configuration re-sults can also be compared with the rough estimatesobtained from the 1-configuration approximation.The rough estimates are shown in Table II inthe column headed by 1E′calc. Better agreementbetween 1Ecalc and 1Eexpt could be obtained byusing more configurations—not higher values of n,but configurations of other types, for example,doubly excited configurations. Notice the goodagreement between (Ei)calc and (Ei)expt.

Discussion

One-electron Sturmian basis functions were in-troduced very early in the history of quantum the-ory, and they are widely used [16 – 23]. Usually theyare defined as a set of radial functions consistingof Laguerre polynomials multiplied by an expo-nentially decaying function, the exponential factorbeing the same for all the members of the basisset. In 1968 Goscinski completed a study [19] inwhich he regarded Sturmian basis functions as be-ing solutions of the Schrödinger equation with a

TABLE IICore ionization energies in Hartrees for the 3-electron isoelectronic series.a

(E0)calc (Ei)calc (Ei)expt 1E′calc 1Ecalc 1Eexpt

Li −7.40869 −5.10052 −5.11080 2.21 2.3082 2.3663Be+ −14.2475 −9.28598 −9.29836 4.88 4.9615 5.0274B2+ −23.3420 −14.7221 −14.7375 8.54 8.6199 8.6909C3+ −34.6893 −21.4086 −21.4290 13.21 13.2807 13.3566N4+ −48.2881 −29.3453 −29.3736 18.87 18.9428 19.0246O5+ −64.1379 −38.5321 −38.5722 25.54 25.6058 25.6995F6+ −82.2359 −48.9689 −48.9906 33.20 33.2670 33.3686Ne7+ −102.587 −60.6558 — 41.87 41.9312 —

a E0 is the ground-state energy, in the 10-configuration approximation, while Ei is the energy of the 3S state obtained by removingan electron from the core of the atom or ion, also in the 10-configuration approximation. The core-ionization energies 1Ecalc =(Ei)calc− (E0)calc, can be compared with experimental values constructed from Moore’s tables [15] and with rough estimates derivedfrom Eqs. (38). These rough estimates are denoted by 1E′calc in Table II.

6 VOL. 79, NO. 1

CORE IONIZATION ENERGIES

weighted 0th-order potential, the weighting factorbeing chosen in such a way that all of the func-tions in the basis set correspond to the same valueof the energy. Goscinski’s way of defining Stur-mian basis functions makes generalization of theconcept very easy. It can be seen that the many-electron basis functions used in the present articleconform to this definition. Such a basis set alwaysobeys a potential-weighted orthonormality relation,the weighting factor being the “basis potential,”V0(x). In the present article and in other publica-tions [3 – 8], we show that when the basis potentialis chosen to be the actual attractive potential dueto the nuclei in an atom or molecule, the resultingmany-electron Sturmian basis set leads to a rapidlyconvergent correlated solution of the many-electronSchrödinger equation of the system, without the useof the self-consistent field approximation. The gen-eralized Sturmian secular equation, which resultsfrom the use of this basis, has several interestingfeatures, which can be seen in Eqs. (25) and (30) ofthe present article. The kinetic energy term has van-ished from the secular equation, and the roots arenot energies, but values of the parameter p0, whichis related to the energy by Eq. (23). Furthermore,in our generalized Sturmian basis, the matrix rep-resenting the nuclear attraction potential is alreadydiagonal, as can be seen from Eq. (30). In conclusion,we feel that the generalized Sturmian method of-fers an interesting and promising alternative to theusual methods used in electronic structure calcula-tions.

References

1. Avery, J. Hyperspherical Harmonics; Applications in Quan-tum Theory; Kluwer Academic Publishers: Dordrecht, 1989.

2. Avery, J.; Herschbach, D. R. Int J Quantum Chem 1992, 42,87.

3. Aquilanti, V.; Avery, J. Chem Phys Lett 1997, 267, 1.4. Avery, J. J Math Chem 1997, 21, 285.5. Avery, J. Adv Quantum Chem 1999, 31, 201.6. Avery, J. Theochem 1999, 458, 1.7. Avery, J.; Sauer, S. Progress in Theoretical Chemistry;

Kluwer Academic Publishers: Dordrecht, to appear.8. Avery, J. Hyperspherical Harmonics and Generalized Stur-

mians; Kluwer Academic Publishers: Dordrecht, 1999.9. Clementi, E. J Chem Phys 1963, 38, 996.

10. Amos, A. T.; Hall, G. G. Proc R Soc London 1961, A263, 483.11. King, H. E.; Stanton, R. E.; Kim, H.; Wyatt, R. E.; Parr, R. G. J

Chem Phys 1967, 47, 1936.12. Löwdin, P. O. Appl Phys 1962, 33(Suppl.), 251.13. Löwdin, P. O. Phys Rev 1955, 97, 1474.

14. McWeeny, R. Methods of Molecular Quantum Mechanics,2nd ed.; Academic Press: New York, 1992; pp. 64–66.

15. Moore, C. E. Atomic Energy Levels; Circular of the NationalBureau of Standards 467; Superintendent of Documents,U.S. Government Printing Office: Washington, DC, 1949.

16. Shull, H.; Löwdin, P.-O. J Chem Phys 1959, 30, 617.17. Rotenberg, M. Ann Phys (New York) 1962, 19, 262.18. Rotenberg, M. Adv At Mol Phys 1970, 6, 233.19. Goscinski, O. Preliminary Research Report No 217; Quan-

tum Chemistry Group, Uppsala University: Uppsala, Swe-den, 1968.

20. Gazeau, J. P.; Maquet, A. Phys Rev A 1979, 20, 727.21. Gazeau, J. P.; Maquet, A. J Chem Phys 1980, 73, 5147.22. Bang, J. M.; Vaagen, J. S. Z Phys A 1980, 297, 223.23. Bang, J. M.; Gareev, F. G.; Pinkston, W. T.; Vaagen, J. S. Phys

Rep 1985, 125, 253–399.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 7