concerning an approximation of the hartree–fock potential by a universal potential function

Concerning an approximation of the Hartree–Fock potential by auniversal potential functionq

R. Gaspar*

Department of Theoretical Physics, Central Research Institute of Physics, Budapest, Hungary

Abstract

One of the most difficult tasks of the many-body problem of atomic physics from the point of view of numerical calculationsis to include the exchange energy. In calculations of statistical atomic physics this energy is taken into consideration with thehelp of a term which is substantially simpler than the corresponding wave-mechanical expression and is related to the totaldensityr of the electrons in the atom. The exchange energy density isga � �4=3�xar1=3

: In a previous work it was shown thatthe reduced effective nucleus chargesZp/Z determined using the ‘self-consistent field’ method disregarding the exchange energycan be described by a universal function independent of atomic number if the quantityx� r=m proportional to the distancerfrom the nucleus is introduced as independent variable. In the present work it is shown that, in the same approach as above andwith the same independent variable, the quantityr1=3

=Z2=3 can also be described by a universal function. With the use of thedensity expression obtainable in this way, the statistical exchange potential can thus be given in a universal form and thenapplied in wave-mechanical calculations. It is expected that the sum of the exchange potential and the electrostatic potentialproposed in the previous work gives a good approximation of the Hartree–Fock potential. Calculations with this potential aremade in order to determine the eigenfunctions and the energies of the electrons of the free Cu atom. The integration of the one-electron Schro¨dinger equation is carried out numerically. The results are reported in Tables 2–10, where, for the ion Cu1, thesolutions of the Fock equations are included as well for comparison purposes. From the data of the tables, it appears clearly thatthe eigenfunctions and eigenvalues calculated using the method proposed here are in good agreement with the eigenfunctionsand energy values determined using the Hartree–Fock method.q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Hartree–Fock method; Universal potential; Exchange energy and potential

1. Introduction

One of the most important tasks of the many-bodyproblem of atomic physics is to determine the electrondistribution of atoms and ions. For heavier atoms thebest tool for this is Fock’s extension of the ‘self-consistent field’ method [1–3]; a comprehensivebibliography is given in Ref. [4]. This method

includes exchange energy as well and constructs theeigenfunction of an atom from one-electron eigen-functions in a determinant form, yielding the rightsymmetry property, namely antisymmetry whenexchanging electrons. The goal of the present workis to show that knowing some results withoutexchange, that is ‘self-consistent field’ results of theHartree kind, a universal potential field can beconstructed for determining one-electron eigenfunc-tions which are good approximations to the solutionsof the Fock equations.

The method will be applied first to the atom Cu andthe results will be compared with those obtained for

Journal of Molecular Structure (Theochem) 501–502 (2000) 1–15

0166-1280/00/$ - see front matterq 2000 Elsevier Science B.V. All rights reserved.PII: S0166-1280(99)00408-X

www.elsevier.nl/locate/theochem

q Presented by P. Gomba´s—Received: 12. XI. 1953. Translatedfrom German by T. Ga´l (Acta Phys. Acad. Sci. Hung. 3 (1954) 263).* Present address: Institute of Theoretical Physics, Kossuth Lajos

University, H-4010 Debrecen, Hungary.

the ion Cu1 by Hartree. Why the Cu atom has beenchosen as the object of this investigation is that it isthe heaviest atom for which the ‘self-consistent field’equations extended to include exchange energy, theso-called Fock equations still have been solved. Thereason the results for the Cu1 ion will be used forcomparison is that only these have been at the author’sdisposal; the method to be described, however, isapplicable to neutral atoms only. As the results willshow, though, the valence electrons (loosely boundelectrons) of the neutral atom do not have too muchinfluence on the electron distribution of the ion; thus aquantitative comparison may also be carried out.Later this issue will be discussed in more detail.

2. The exchange energy and its potential

First, it will be pointed out briefly that, similar towhat Lenz, Jensen and Gomba´s did [5–7], a potentialcan be given for the exchange energy which was intro-duced into the statistical theory of atoms by Diracearlier [8]. Recently, Slater has also proposed anexchange potential [9], which for free electrons, hasthe same form as Dirac’s, simply one and a half of it.Since these potentials play an essential role in thepresent study, here, a short review of the introductionof the Dirac potential and its important properties willbe given first.

Given an atom with electron densityr and an elec-tron, belonging to this atom, with densityn of thewave-mechanical sense, the change in the exchangeenergy due to removing the electron cloud of densityn from the atom is

DEa � xaZr4=3 dv 2

Z�r 2 n�4=3 dv 2

Zn4=3 dv

� �;

�1�where dv denotes the volume element and the integralis taken over the whole space, andxa � �3=4� ��3=p�1=3e2

: In Eq. (1) the single terms can be inter-preted very well. The first term is the exchange energyof the atom, the second is the exchange energy of thepositive ion with density r 2 n remaining afterremoving the electron and, finally, the third is theself-exchange energy of the electron. Sincer $ nholds everywhere, the second term of Eq. (1) can betransformed with the help of the binomial series to get,

neglecting higher-order small terms,

DEa � xa43

Zr1=3n dv 2

Zn4=3 dv

� �: �2�

If n � ucu 2; c being the eigenfunction of the electron

in question, normalized to 1, the first term of Eq. (2) isthe wave-mechanical mean value of the potentialenergy

ga � 43xar

1=3 �3�

and the second term is the self-exchange energy of theelectron with density distributionn . Thus, leaving thesecond term of Eq. (2), in the remaining term the self-exchange energy of the electron with density distri-butionn is to be included as well. Hence, when settingup the Schro¨dinger equation with the use of the poten-tial energy (3), the self-exchange energy needs to becompensated in some way. The potential correspond-ing to the potential energy (3) is the Dirac potential or,multiplied by one and a half, the Slater potential.

3. The universal potential

In one of the author’s previous works [10] it wasalready shown that the reduced effective nucleuscharges of neutral atoms can be written in the form

Zp

Z� e2l0x

1 1 A0x�4�

as a good approximation. With the help of thisequation the effective potential

V � Zpe

r�5�

can be obtained immediately for all atoms. In Eq. (4)

l0 � 0:1837; A0 � 1:05 �6�and

x� rm; �7a�

while

m � 0:8853a0

Z1=3 �7b�

is the length unit known well from statistical atomicphysics, changing from atom to atom. About the

R. Gaspar / Journal of Molecular Structure (Theochem) 501–502 (2000) 1–152

degree of validity of Eq. (4), Fig. 1 of the author’sabove-mentioned work [10] gives insight.

In the statistical theory disregarding exchange (theThomas–Fermi approach) the densities of neutralatoms can also be transformed into a frame of refer-ence in which they show a universal form. The mostsuitable is to take the quantityx defined by Eq. (7a) asabscissa and�r=Z2�1=3 as ordinate. Here, it will beexamined to what extent this universal behaviour ofthe density curves holds when making more accuratewave-mechanical calculations.

In Figs. 1–3 (made with the help of the calculationsreported in Refs. [11–14]) the quantity�r=Z2�1=3obtained using the ‘self-consistent field’ methoddisregarding the exchange term is plotted as functionof 1/x and x. In the figures, with broken line, the

density distribution obtained from the Thomas–Fermi theory is also displayed, for which

r

Z2

� �1=3

� 14pm3Z

w0

x

� �1=2

: �8�

In Eq. (8) w0 is the solution of the Thomas–Fermiequation for neutral atoms thus, it is a universal func-tion, independent of atomic number, whilem 3Z is auniversal constant.

From Fig. 1 it can be clearly seen that, apart fromthe regions near the nuclei (large 1/x values), over aconsiderable interval, the density distributions ofneutral atoms are only weakly dependent on atomicnumber in this frame of reference. The curve plottedwith continuous line in the figure can be described by

R. Gaspar / Journal of Molecular Structure (Theochem) 501–502 (2000) 1–15 3

Fig. 1. The reduced electron densities of neutral atoms in universal system of reference. To get a better view on the relations near the nucleus 1/xis introduced as abscissa. The arrows at the bottom indicate the positions of maxima of the radial eigenfunctions of the 1s electrons of the singleatoms. The points, marked by different symbols, are determined by the densities calculated with the ‘self-consistent field’ method withoutexchange. The curve with the broken line is the analytical curve�r 0=Z2�1=3 � C e2ax

=�1 1 Ax� with the parameters (10).


Fig. 2. The reduced electron densities of neutral atoms in universal system of reference. A part of Fig. 1 in a larger scale and with more pointscalculated with the ‘self-consistent field’ method. The meanings of the single symbols are as given in Fig. 1.

Fig. 3. The reduced electron densities of neutral atoms at large distance from the nucleus in universal system of reference. Accordingly,x istaken as abscissa. The symbols are as in Fig. 1.

the equation

r 0

Z2

!1=3

� C e2ax

1 1 Ax�9�

where the constants have the following values:

C � 3:1a210 ; a � 0:04 and A� 9: �10�

The constantC can be determined more accuratelyin the following way. The density has to be normal-ized, which means, since considering neutral atoms,

4pZ∞

0r 0r2 dr � Z �a�

Substituting Eq. (9) into this equation and changing tothe variablex with the help of Eq. (7), after simplifi-cation

4pm3ZC3Z∞

0

e23ax

�1 1 Ax�3 x2 dx� 1 �b�

can be obtained.Considering thatm3Z � 0:88533a3

0; Eq. (b), thusindependent of atomic number, makes it possible todetermineC independent of atomic number. Utilizingnow thatZ∞

0

e23ax

�1 1 Ax�3 x2 dx� 2e3a=A

A3 Ei 23aA

� �

� 1 16aA

19a2

2A2

!

23

2A3 1 1a

A

� �;

C ù 3:2a210 arises, which is in quite good agreement

with the valueC � 3:1a210 given earlier. However, the

curve of r 0 gives good local mean values with thelatter value ofC in the important regions; thereforethis value is kept.

In Fig. 2 a part of Fig. 1 which belongs to smallvalues of 1/x is plotted in a bigger scale and, also,more points are included in the plot. In this regionthe differences between the values ofr correspondingto the same coordinatex are considerably smaller. Itcan be clearly seen from the two figures that thedensity values tend to constant values as 1=x! ∞:

This is especially striking for Be because in the caseof this element this asymptotic behaviour appears

already at small 1/x values. The values of theconstants, however, vary from atom to atom andgenerally increase with atomic number. The curveof the Thomas–Fermi theory is a universal curveand it is obvious that in the case of heavier atoms itcan take this steadily increasing value only if it, itself,tends to infinity as 1=x! ∞: Simple considerationsshow that this can be the case only in the order of(1/x)3/2.

Fig. 3 exhibits the changes of the density distri-butions far from the nucleus. It can be seen that,though the dispersion of the density values is quitelarge, dependence on atomic number can be dis-regarded in a first step, and they can be approximated,for a purpose to be detailed later, by the curve plottedwith continuous line. The Thomas–Fermi density, onthe other hand, goes apparently above the wave-mechanical density curve in this outer region andtends to zero only very slowly [11–14].

Comparing the three figures, it can be concludedthat the densities determined with the ‘self-consistentfield’ method without exchange can be described wellin general with the function (9) in the chosen specialframe of reference apart from the regions near thenuclei and the outer regions. Further, in the regionsfar from the nuclei, the behaviour of the function (9) isin accordance with the behaviours of the wave-mechanical density distributions since it vanishesexponentially. In order to have a better view onthe extent of the region which can be considered,the sites where the maxima of the radial densitiesfor the most inner, 1s electrons of the singleelements are marked in Fig. 1 (an arrow and thechemical symbol of the element considered, togetherwith the symbol used in the plot). It is worth noting,further, that in the case of the element Cu, for whichthe calculations are done in detail, the maximum ofthe radial density of the 3d electron, that is the mostouter non-valence electron, is atxm ù 2:16; near towhich the approximating function (9) approachesthe values obtained using the ‘self-consistent field’method still well.

With the use of Eqs. (5), (3) and (9) the universalpotential field

V 0 � Zpe

r1

43xaer 01=3 � Ze

re2l0x

1 1 A0x1

C 0

ee2ax

1 1 Ax

�11�


arises, wherel0, A0, a andA take the values given byEqs. (6) and (10) and

C 0 � 43xaZ2=3C: �12�

A very pleasant property of the potential (11) isbeing calculable for any element without any specialdifficulty by simply changing the atomic number.Such transformation, however, has to be made withcare for the change ofx as well because of the depen-dence ofm on the atomic number, according to Eq.(7). The potential (11) takes the exchange energy intoaccount as well. As mentioned already, the exchangeenergy contains also the so-called self-exchangeenergy of the electron under consideration, in a waythat it is hard to separate. On the other hand, as hasalready been pointed out in detail in the author’sprevious work [10], in the potential (5) the electro-static self-potential of the electron considered isincluded. The two self-potentials have oppositesigns and compensate each other just in the rightway in the potential (11). Hence, the potential (11)gives a very good approach to the real potential fieldof the electron considered.

Also, it has to be mentioned that, near the nucleus,the second term of Eq. (11), the exchange term, givesonly a small correction compared to the first termtherefore, near the nucleus, the difference betweenthe real density and the one defined by the function(9) is irrelevant for the potential (11). In the outerregions, far from the nucleus, the dispersion of thereal density distribution is relatively large. However,only relatively small parts of the eigenfunctions, overthe maxima, correspond to these regions, even in thecase of the very loosely bound 3d electron, so therough average of the density distribution [Eq. (9)]can be sufficient here, too.

4. The Schrodinger equation and its solution

With the use of the potential (11) the Schro¨dingerequation for an electron in a neutral atom is

2h2

8p2m0Dc 2

Zpe2

r1

43xar

01=3 !

c � Ec: �13�

Here, as usual,m0, h, E andc denote the mass of theelectron, Planck’s constant, the energy of the electron

and its eigenfunction, respectively. Since the potential(11) is spherically symmetric the solution of Eq. (13)can be assumed in the form

c � f �r�r

Ylm�q;w�; �14�

whereYlm is the spherical function corresponding tothe azimuthall and magneticm quantum numbers.f(r) is the radial eigenfunction, which satisfies theequation

d2f

dr2 1

"8p2m0

h2

E 1

Zpe2

r1

43xar

1=3

!

2l�l 1 1�

r2

#f � 0

�l � 0;1;2;…�:

�15�

Eq. (15) has to be solved with the following boundaryconditions:

f �0� � 0;

f 0�0� � arbitrary constant;

limr!∞ f �r� � 0:

�16�

A solution of Eq. (15) which satisfies the boundaryconditions (16) exists only for certain values of theenergy parameter. Let these values be denoted byEnl

here. On grounds of the foregoing, substitutingZp

from Eq. (4) andr 1/3 from Eq. (9) into Eq. (15),differential equations are obtained, the solutions ofwhich approach the solutions of the Fock equationswell. The equations (15) can be set up straight awayfor elements of any atomic number so they have notonly the useful properties mentioned in the previoussection but the potential contained in them can beproduced for every atom (for any atomic number)simply by changing two factors when changing tovariablex when integrating. If this is not done, thena coordinate scaling has to be performed, of course.

The radial Schro¨dinger equation has been solvednumerically since the solution could not have beenobtained analytically. Also, numerical integrationhas been required in order to make a better com-parison of the eigenfunctions with the solutions ofthe Fock equations possible. In the case of numericalapproaches one has to be content with less good


approximations for the eigenfunctions. The numericalintegration has been carried out on the equation

d2f

dx2 1 e 1g

xe2l0x

1 1 A0x1 z

e2ax

1 1 Ax2

l�l 1 1�x2

" #f

� 0; (17)

which differs from Eq. (15) in being changed over toatomic units,1 the variablex defined by Eqs. (7a) and(7b) being introduced instead ofr and the notation

e � 2Em2e22a210 ;

g � 2Zma210 and z � 8

3xaC0:88532e22a0

�18�

being applied. The boundary conditions (16) are validbasically in the original form after introducing thevariable x, proportional tor. With the help of thefirst two boundary conditions and Eq. (17), the func-tion f can be expanded into a Taylor series aroundx�0; that is

f �x� � u�x�xl11; �19�

whereu(x) has the following form:

u�x� � u�0�1u0�0�

1!x 1

u00�0�2!

x2 1u000�0�

3!x3

1u�4��0�

4!x4 1 …:

The values of the functionu(x) and its derivatives atx� 0 can be determined easily with the use of Eqs.(16) and (17).

Thus, with the help of Eq. (19), the values of thefunction f can be calculated nearx� 0:

The differential equation to be solved now beingavailable, some words may be said about the methodused for solving it. If starting with the help of Eq. (19)when integrating numerically, the eigenfunction auto-matically fulfills the first two boundary conditions butnot the third one. Since the solution has been carriedout numerically, having an eigenvalue-problem, the

differential equation (17) had to be integrated numer-ically for different values ofe until the third of theboundary conditions (16) was fulfilled as well. Carry-ing out the integration two types of solutions arise.One of them crosses the abscissa while the othertends to^∞ with increasingx values after reachinga minimum. Obviously, the eigenvalue, the right valueof e , belongs to a curve which, asymptotically,approaches the axisx with x! ∞: Thus, the rightvalue of e lies between thee values correspondingto the two curves mentioned above. Since in practicethe exact eigenvalue, so also the eigenfunction, cannotbe produced, in general the following procedure hasbeen followed. In order to obtain accuracy, as manyintegrations have been applied so that the last decimaldigit of the value ofe can be delimited with the helpof the two aforementioned curve types. It is knownthat, in general, the eigenvalue parameter is moreaccurate, with at least one order, than the eigenfunc-tion of the approximation. This fact is advantageousfor many quantum mechanical approximate calcula-tions because a rougher eigenfunction suffices as wellfor determining the eigenvalue parameter. For exam-ple, in this way good energy calculations can becarried out with the variational method even whenusing simpler analytical forms for the eigenfunctions.In the present case, where determining the eigenfunc-tions is the primary goal, the unpleasant side of thisfact shows. Since, if one wants the eigenfunction to bedetermined with a proper accuracy (which, in thepresent case, has been taken to be the usual accu-racy of ‘self-consistent field’ calculations (seetables)), the eigenvalue parametere has to bedetermined with much greater accuracy, whichincreases the number of the integrations neededconsiderably. Generally, in order to reach the requiredaccuracy and get the right eigenfunctions for largexvalues as well,e has had to be determined with accu-racy of five digits, in several cases, or, in some cases,even six digits.

Considering that no other calculation apparatus hasbeen available than just the ordinary multiplicationmachines, a possibly simple method has been chosento solve the differential equation. The used method isa version of the Adams–Sto¨rmer method which hasalready been applied and described in detail byProkofjeff [15]. An important advantage of thismethod is that its extrapolation formula can be


1 The atomic units are the following: the charge unite, the posi-tive elementary charge; the mass unitm0, the mass of the electron;the length unita0, the smallest Bohr Hydrogen-radius; the units ofother quantities can be derived from these, e.g. the energy unit ise2/a0, the double of the ionization energy of the atomH.

given in the very simple form

fn11 � 2fn 2 fn21

1 h2 jn 1112

D 2n j 1 D 3

nj 11920

D 4nj

� �� ;

�20�where

j � 2f e 1g

xe2l0x

1 1 A0x1 z

e2ax

1 1 Ax2

l�l 1 1�x2

" #�21�

in the present case andD2j; D 3j andD 4j denote thesecond, the third and the fourth differences of thequantityj , respectively, andh is the interval length.In the present work the interval length has been takenso small that the fourth difference gives already negli-gibly little correction. This way the steps taken havebeen managed to be controlled without the need forcorrecting the results step by step. In this work, wherethe numerical calculations have been carried out forthe atom Cu,h has been chosen as listed in Table 1.

The most outer extremum of the radial eigenfunc-tion 4s lies atx ù 6:62 approximately and that of theeigenfunction 3d atx ù 2:16: As can be seen, in theregion containing the maxima the steps are still quitesmall, and the first of the bigger steps has been takenafter reaching the last maximum, where the shapes ofthe eigenfunctions are already flat enough. A moreprecise way of controlling is checking the resultafter each extrapolation step with the formula

fn � 2fn21 2 fn22 1 h2 jn 2 D 1n j 1

112

D 2nj

� ��22�

and when necessary, correcting the steps alreadymade [15]. In several cases this more complicatedmethod has been applied as well, this way beingconvinced several times about the rightness of theresults obtained with the method followed generally.

5. Results and discussion

The calculations have been carried out for the atomCu. The reason for choosing this atom is that this is theheaviest atom for which ‘self-consistent field’ calcu-lations have been done considering exchange energy.Solutions of the ‘self-consistent field’ Fock equations,suitable for comparison, however, have been availableonly for the Cu1 ion therefore, with no other option,these have to be given. If one wishes to estimate the


Table 1About the regions of application of the interval lengthsh used forthe numerical integration of the differential Eq. (17)

Start End h

0.000 0.090 0.0060.090 0.288 0.0180.288 0.936 0.0360.936 2.160 0.0722.160 4.320 0.1444.320 10.368 0.28810.368 – 0.576

Table 2The radial eigenfunctions of the 1s electrons of the Cu atom and theCu1 ion calculated with the universal potential field, and those fromthe Fock equations. The presented values are given in atomic units

r Hartree–Fock Present

0.000 0.000 0.0000.005 1.328 1.3410.010 2.299 2.2930.015 2.985 2.9910.020 3.445 3.4590.025 3.729 3.7520.030 3.876 3.8890.035 3.918 3.9290.040 3.881 3.8890.050 3.645 3.6500.060 3.290 3.2900.070 2.890 2.8850.080 2.488 2.4800.090 2.110 2.1030.100 1.769 1.7600.120 1.211 1.2010.140 0.809 0.7980.160 0.530 0.5210.180 0.344 0.3350.200 0.221 0.2130.220 0.141 0.1360.240 0.089 0.0860.260 0.056 0.0560.280 0.036 0.0340.300 0.023 0.0180.350 0.007 0.0020.400 0.002 –0.450 0.001 –

influence of the most outer valence electron on theeigenfunctions it can be said as a rough approach thatthe valence electron shields the nucleus in part and, inaccordance with this shielding effect, the effectivenucleus charge decreases in comparison with the effec-tive nucleus charge of the ion. The expression used herefor the exchange energy works against this effect. Thisenergy term is proportional to the 1/3 power of thedensity and, because of this dependence, in the case of

small densities this term overcompensates the faultmentioned above. Thus, it is understandable that theeigenfunctions to the shapes of which the parts of theeffective potential being far from the nucleus are rele-vant contract a little towards the nucleus. In the energythis effectmanifests itself insucha way that the energy issomewhat larger, considering its absolute value, thanthe energy of the corresponding electron of the Cu1 ion.




0.000 0.000 0.0000.005 0.407 0.3960.010 0.700 0.6900.015 0.896 0.8950.020 1.011 1.0100.025 1.061 1.0580.030 1.057 1.0520.035 1.009 1.0070.040 0.926 0.9170.050 0.686 0.6720.060 0.384 0.3660.070 0.054 0.0340.080 20.280 20.3000.090 20.602 20.6360.100 20.900 20.9230.120 21.403 21.4200.140 21.770 21.7780.160 22.007 22.0140.180 22.132 22.1330.200 22.168 22.1640.220 22.134 22.1320.240 22.050 22.0430.260 21.932 21.9320.280 21.795 21.7900.300 21.646 21.6420.350 21.270 21.2670.400 20.940 20.9380.450 20.674 20.6720.500 20.472 20.4690.550 20.328 20.3170.600 20.225 20.2050.700 20.104 20.1200.800 20.048 20.0960.900 20.022 20.0661.000 20.011 20.0381.100 20.004 20.0021.200 20.022 –1.300 20.001 –

Table 4The radial eigenfunctions of the 2p electrons of the Cu atom and theCu1 ion calculated with the universal potential field, and those fromthe Fock equations. The presented values are given in atomic units


0.000 0.000 0.0000.005 0.016 0.0210.010 0.061 0.0600.015 0.128 0.1290.020 0.213 0.2210.025 0.310 0.3210.030 0.416 0.4370.035 0.528 0.5490.040 0.644 0.6550.050 0.877 0.9010.060 1.103 1.1190.070 1.312 1.3230.080 1.500 1.5080.090 1.663 1.6690.100 1.801 1.8290.120 2.002 2.0030.140 2.111 2.1250.160 2.142 2.1450.180 2.113 2.1040.200 2.040 2.0270.220 1.934 1.9190.240 1.808 1.7910.260 1.670 1.6570.280 1.528 1.5100.300 1.386 1.3640.350 1.055 1.0500.400 0.778 0.7750.450 0.561 0.5630.500 0.398 0.3980.550 0.279 0.2800.600 0.195 0.1930.700 0.093 0.0910.800 0.045 0.0420.900 0.022 0.0181.000 0.011 0.0051.100 0.005 –1.200 0.002 –1.300 0.001 –1.400 0.000 –




0.000 0.000 0.0000.005 0.152 0.1540.010 0.261 0.2640.015 0.334 0.3370.020 0.376 0.3800.025 0.393 0.3970.030 0.389 0.3920.035 0.368 0.3710.040 0.334 0.3360.050 0.238 0.2370.060 0.119 0.1150.070 20.010 20.0160.080 20.139 20.1470.090 20.260 20.2690.100 20.370 20.3800.120 20.544 20.5440.140 20.651 20.6590.160 20.692 20.6980.180 20.677 20.6800.200 20.614 20.6160.220 20.515 20.5150.240 20.390 20.3880.260 20.247 20.2450.280 20.094 20.0930.300 0.062 0.0630.350 0.438 0.4360.400 0.757 0.7540.450 1.001 0.9980.500 1.170 1.1640.550 1.268 1.2690.600 1.311 1.3140.700 1.275 1.3050.800 1.147 1.1550.900 0.983 0.9881.000 0.817 0.8171.100 0.663 0.6591.200 0.531 0.5181.300 0.420 0.4071.400 0.329 0.3151.600 0.198 0.1841.800 0.117 0.1052.000 0.069 0.0592.200 0.040 0.0342.400 0.023 0.0202.600 0.014 0.0142.800 0.008 –3.000 0.004 –3.200 0.002 –3.400 0.001 –3.600 0.000 –

Table 6The radial eigenfunctions of the 3p electrons of the Cu atom and theCu1 ion calculated with the universal potential field, and those fromthe Fock equations. The presented values are given in atomic units


0.005 0.006 0.0060.010 0.022 0.0230.015 0.047 0.0490.020 0.078 0.0810.025 0.113 0.1180.030 0.152 0.1580.035 0.193 0.1990.040 0.234 0.2420.050 0.317 0.3270.060 0.395 0.4070.070 0.465 0.4770.080 0.524 0.5300.090 0.572 0.5850.100 0.601 0.6190.120 0.642 0.6510.140 0.632 0.6420.160 0.583 0.5830.180 0.503 0.4990.200 0.400 0.3920.220 0.281 0.2700.240 0.152 0.1390.260 0.018 0.0690.280 20.116 20.1330.300 20.249 20.2660.350 20.554 20.5700.400 20.804 20.8190.450 20.993 21.0070.500 21.123 21.1360.550 21.200 21.2130.600 21.235 21.2520.700 21.212 21.2240.800 21.115 21.1240.900 20.986 20.9881.000 20.848 20.8431.100 20.717 20.7041.200 20.598 20.5361.300 20.494 20.4711.400 20.405 20.3791.600 20.269 20.2411.800 20.176 20.1502.000 20.114 20.0922.200 20.073 20.0562.400 20.047 20.0342.600 20.030 20.0212.800 20.019 20.0133.000 20.012 20.0083.200 20.007 20.0063.400 20.004 –3.600 20.002 –3.800 20.001 –

The radial eigenfunctions obtained by integratingthe radial equations of the Cu atom can be found inTables 2–8. In these tables the solutions of the Fockequations for the Cu1 ion are also given.

It is without any doubt that the universal potentialfield used in the calculation gives a good approxi-mation of the real potential only within certain limits.It must be examined, thus, in what approach it isapplicable and what kind of errors should be takencare of. One of the most striking faults is that whilethe potential of a singly charged positive ion ise/r atlarge distances from the nucleus, that is it decreasesonly very slowly with distance, the universal potentialfield vanishes very quickly, exponentially. In Fig. 4the potential energy of an electron is plotted, dis-regarding a proportionality factor (2m 2), in thefollowing potential fields: (1) in the field of a pointcharge of magnitudee located at the place of thenucleus; (2) in the potential field (5) ifZ � 29; (3)in the potential field (11) ifZ � 29:

From the figure it can be seen that the curves 1 and3 go close to each other in quite a big interval, fromaboutx1 < 9 to x2 < 14; and intersect atx < 10:5:Thus, even in the outer parts of the atom, the universalpotential (11) gives a quite good approximation. Ofcourse, the same cannot be said about the curve 2.Since, this curve crosses the curve 1 already atx ù6:6 and, after that, its value decreases quickly. Ofcourse, in the inner regions of the atom there doesnot have to be any accordance between the two poten-tials since one of them is the potential of a pointcharge while in the case of the other the effectivenucleus charge increases approaching the nucleus. Itcan be expected for the atom Cu that the fault of theuniversal potential mentioned above influencesmostly the most outer electrons, i.e. the electrons


Table 7The radial eigenfunctions of the 3d electrons of the Cu atom and theCu1 ion calculated with the universal potential field, and those fromthe Fock equations. The presented values are given in atomic units


0.000 0.000 0.0000.005 0.000 0.0000.010 0.000 0.0000.015 0.001 0.0010.020 0.002 0.0020.025 0.003 0.0030.030 0.005 0.0050.035 0.008 0.0080.040 0.011 0.0110.050 0.019 0.0200.060 0.030 0.0320.070 0.044 0.0470.080 0.061 0.0640.090 0.080 0.0850.100 0.101 0.1070.120 0.149 0.1570.140 0.203 0.2130.160 0.261 0.2730.180 0.321 0.3350.200 0.382 0.3980.220 0.442 0.4600.240 0.501 0.5210.260 0.557 0.5800.280 0.610 0.6340.300 0.660 0.6870.350 0.770 0.8000.400 0.855 0.8920.450 0.917 0.9590.500 0.958 1.0050.550 0.981 1.0330.600 0.991 1.0450.700 0.978 1.0340.800 0.937 0.9890.900 0.882 0.9241.000 0.821 0.8501.100 0.759 0.7731.200 0.698 0.6961.300 0.639 0.6231.400 0.585 0.5541.600 0.488 0.4341.800 0.406 0.3372.000 0.337 0.2592.200 0.279 0.1992.400 0.231 0.1532.600 0.191 0.1172.800 0.158 0.0903.00 0.130 0.0693.200 0.107 0.0533.400 0.088 0.0413.600 0.072 0.032

Table 7 (continued)


3.800 0.059 0.0254.000 0.049 0.0214.500 0.029 0.0145.000 0.018 0.0085.500 0.010 0.0036.000 0.006 0.0017.000 0.002 –8.000 0.001 –

having the most extensive electron clouds, namely the4s and the 3d electrons. Studying the radial densitydistributions of the single electrons (the datacontained by Tables 2–8 must be squared then multi-plied by 4p), it can be found that, with a 90% prob-ability, the loosely bound electrons are within thespheres of radiusx < 8 (3d electron) andx < 18 (4selectron). From this the conclusion can be drawn that,except for the 4s electron, the behaviour of the elec-trons is determined mostly by the potential field domi-nating within the sphere of radiusx < 14: Within thissphere, however, the universal potential is a goodapproximation of the real one where the incorrectfall of the curves appearing at large values ofr doesnot take effect yet. Hence, as can be expected as well,the incorrect asymptotic behaviour of the potentialfunction has only small influence on the eigenfunc-tions and the energy eigenvalues. Even in the case ofthe 4s electron, for which the electron-cloud spreadsover a very large area, over a substantial region themotion of the electron can be examined with a correctpotential field.

Comparing the eigenfunctions calculated using theFock approach of the ‘self-consistent field’ methodwith the ones calculated with the universal potentialfunction shows that the 1s, the 2s, the 2p, the 3s andthe 3p eigenfunctions almost completely coincide.But the eigenfunction of the 3d electron calculatedwith the universal potential field is a little flattenednear the nucleus compared with the eigenfunction ofthe Fock kind. For the outermost electron, the 4s elec-tron, there is no basis for comparison because theFock equations of the ‘self-consistent field’ calcula-tions can be given without difficulty only for atoms orions with closed electron structure, thus the calcula-tions have been carried out also only for the ion Cu1.



r Present

0.000 0.0000.005 0.0490.010 0.0840.015 0.1080.020 0.1210.025 0.1260.030 0.1240.035 0.1180.040 0.1060.050 0.0750.060 0.0360.070 20.0060.080 20.0480.090 20.0870.100 20.1220.120 20.1770.140 20.2090.160 20.2190.180 20.2120.200 20.1870.220 20.1520.240 20.1090.260 20.0600.280 20.0120.300 0.0420.350 0.1600.400 0.2580.450 0.3220.500 0.3560.550 0.3590.600 0.3400.700 0.2470.800 0.1130.900 20.0361.000 20.1821.100 20.3151.200 20.4331.300 20.5241.400 20.6001.600 20.6961.800 20.7402.000 20.7342.200 20.7052.400 20.6592.600 20.6062.800 20.5493.000 20.4913.200 20.4373.400 20.3853.600 20.338

Table 8 (continued)

r Present

3.800 20.2944.000 20.2564.500 20.1785.000 20.1225.500 20.0836.000 20.0567.000 20.0248.000 20.008

In this case comparison with experience can be madeonly by comparing the energy values.

In Table 9 the full radial density of the Cu1 ion isgiven in three cases, namely, when calculated usingHartree’s method, Fock’s method and the universalpotential. From the table it can be seen well that thedensity calculated with the universal potential is moreconcentrated in the area near the nucleus than thecorresponding density calculated using the Fockequations, which it runs close to, though, the densitycalculated with Hartree’s method extends more, that ishas considerable magnitude even at bigger values ofr.The radial density calculated with the use of theuniversal potential, thus, gives a quite good approx-imation of the density calculated with the Hartree–Fock method.

It was mentioned at the start of this work that twoforms had been proposed for the exchange potentialwhich differ only by a factor 3/2. If the Dirac potentialdefined with the formula (3) isga, Slater’s potential


Table 9The full radial densities of the Cu1 ion calculated in the Hartree andthe Hartree–Fock approaches of the ‘self-consistent field’ methods,as well as that calculated with the universal potential. The valuespresented in the table are given in atomic units

r Hartree Hartree–Fock Present

0.000 0.00 0.00 0.000.005 3.9 3.9 3.960.010 11.7 11.7 11.630.015 19.7 19.7 19.840.020 26.3 26.3 26.590.025 31.0 31.0 31.410.030 33.7 33.7 34.060.035 34.9 34.9 35.220.040 34.8 34.8 35.170.050 32.7 32.8 33.180.060 30.0 30.2 30.460.070 28.1 28.3 28.540.080 27.4 27.7 27.890.090 27.9 28.3 29.640.100 29.4 29.9 30.670.120 33.61 34.20 34.390.140 37.35 37.96 38.470.160 39.32 39.83 40.010.180 39.23 39.59 39.420.200 37.36 37.62 37.380.220 34.35 34.53 34.300.240 30.81 30.97 30.740.260 27.24 27.43 27.460.280 23.98 24.26 24.230.300 21.26 21.67 20.710.350 17.19 18.04 18.550.400 16.48 17.72 18.480.450 17.61 19.13 20.080.500 19.22 20.89 21.940.550 20.53 22.17 23.390.600 21.16 22.74 24.090.700 20.45 21.70 23.170.800 18.02 18.89 20.060.900 15.02 15.54 16.361.000 12.17 12.39 12.831.100 9.72 9.72 9.821.200 7.75 7.57 7.101.300 6.12 5.91 5.541.400 5.01 4.62 4.131.600 3.37 2.89 2.301.800 2.35 1.86 1.292.000 1.69 1.21 0.732.200 1.24 0.81 0.422.400 0.91 0.55 0.242.600 0.67 0.37 0.142.800 0.50 0.25 0.083.000 0.37 0.17 0.043.200 0.28 0.11 0.033.400 0.20 0.08 0.02

Table 9 (continued)

r Hartree Hartree–Fock Present

3.600 0.15 0.05 0.013.800 0.11 0.03 0.014.000 0.08 0.02 0.004.500 0.03 0.00 –5.000 0.01 0.00 –5.500 0.00 0.00 –

Table 10The energy levels of the electrons of the Cu atom calculated withboth methods described in detail here (Slater, Dirac). For compar-ison the corresponding results of the ‘self-consistent field’ methodin the Hartree and the Hartree–Fock approach are also given. In thecase of the 4s electron, for which results of the ‘self-consistent field’method are not available, the first ionization energy determinedexperimentally should be considered, which is 7.68 eV, i.e. inatomic units, 0.2824e2/a0. The energy values reported in the Tableare given in atomic units, i.e. in unitse2

=a0 � 27:20 eV

nl Hartree Hartree–Fock Slater Dirac

1s 2329 2329.2 2334.0 2328.22s 239.225 241.15 240.66 238.772p 234.93 235.915 236.49 234.243s 24.493 25.3255 25.675 24.9463p 23.039 23.6395 24.205 23.4833d 20.5975 20.8065 21.513 20.89784s – – 20.5921 20.3784

can be written asgS � �3=2�ga: The calculations havebeen carried out not only with the Dirac potential butalso with Slater’s one and the results obtained for theenergy eigenvalue are summed up in Table 10. It canbe seen well that the energy eigenvalues calculatedwith Slater’s potential are generally smaller than theones determined with the Hartree–Fock method and,especially for the loose bound electrons, very bigdifferences can be found. The energy eigenvaluescalculated with the Dirac potential, apart from theenergies of the 2s and the 2p electrons, fall close tothe Hartree–Fock eigenvalues and decent results havebeen obtained even for the loosely bound electrons.The case of the 2s and the 2p electrons is being dealtwith individually. Studying the approximate effectivepotential (4) in more detail (see Table 1 in the work[10]) it can be seen that still significant parts of the 2sand the 2p eigenfunctions fall into the region fromr ù0:4 to r ù 1:2 and in this region the analytical butapproximate effective nucleus charge (4) is smallerthan the one determined using Hartree’s method.This has the consequence that the mean value of thepotential energy increases and, also, the energy eigen-value is greater than expected. Hence, it is probablethat if a changed effective nucleus charge determinedusing Hartree’s method was taken as basis for thecalculations instead of the analytical form (4), theagreement of the energy eigenvalues (and, alongwith them, also the eigenfunctions, of course) withthe corresponding results of the Hartree–Fock method

would be better in the cases of the 2s and the 2pelectrons as well. It can also be seen well fromTable 10 that the different behaviour of the 2s and2p eigenvalues is not due to the difference in theform of the exchange potential since, in both theDirac and Slater series, the eigenvalues of these elec-trons behave differently from the other elements of theseries, comparing them with the Hartree–Fock eigen-values, and also the directions of the differences arethe same in both series.

About the method described in the present work itcan be said in general that although the calculations donot have the character of methods of ‘self-consistentfield’, the results give a good approach of the eigen-functions and the eigenvalues determined using theHartree–Fock method. In constructing the effectivepotential taking the exchange energy into considera-tion has proved to be relevant, the self-exchange partof which compensating the electrostatic self-energybeing in the potential (4), making it possible that theeffective potential given this way gives a goodapproximation of the real potential also for the looselybound electrons. From the exchange energies Dirac’sform was chosen here, and choosing the Dirac poten-tial has been verified by the results.

The calculations are notably simpler than thoseusing methods of ‘self-consistent field’. By virtue ofthe simple analytical form of the potential the methodis particularly suitable for implementation with biggercalculators since just simple changes of factors have to


Fig. 4. The potential energy of an electron in the Cu atom. (1)22me2=x �2e2

=mx is the energy of the electron if the Cu1 core is compressed into apoint charge). (2)22me2V (V is the potential of the Cu atom as in Eq. (5)). (3)22me2V 0 (V0 is the potential of the Cu atom as in Eq. (11), that isincluding exchange). For a better view a quantity proportional to the potential energy is plotted. The proportion factor is 2m 2.

be made for elements of different atomic number.Finally, it can also be mentioned that, applying thepotential, considerable reduction in calculationalwork can be expected also in molecule and crystaltheoretical calculations.

Also, here the author thanks Ms B. Molna´r, Mr B.Molnar and Mr A. Szabo´ for carrying out the numer-ical calculations.

References

[1] V. Fock, Z. Phys. 61 (1930) 126.[2] V. Fock, Z. Phys. 62 (1930) 795.[3] J.C. Slater, Phys. Rev. 35 (1930) 210.

[4] P. Gomba´s, Theorie und Lo¨sungsmethoden des Mehrteilchen-problems der Wellenmechanik, Birkha¨user, Basel, 1950.

[5] W. Lenz, Z. Phys. 77 (1932) 713.[6] H. Jensen, Z. Phys. 77 (1932) 722.[7] P. Gombas, Z. Phys. 121 (1943) 523.[8] P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26 (1930) 376.[9] J.C. Slater, Phys. Rev. 81 (1951) 385.

[10] R. Gaspar, Acta Phys. Hung. II (1952) 151.[11] D.R. Hartree, W. Hartree, Proc. Roy. Soc. (A) 149 (1935) 210.[12] D.R. Hartree, W. Hartree, Proc. Roy. Soc. (A) 166 (1938) 450.[13] M.F. Manning, L. Goldberg, Phys. Rev. 53 (2) (1938) 622.[14] M.F. Manning, J. Millmann, Phys. Rev. 49 (2) (1936) 848.[15] E. Kamke, 3, Differentialgleichungen, Lo¨sungsmethoden und

Losungen, 1, Akademische Verlagsgesellschaft Becher&ErlerKom-Ges, Leipzig, 1944, p. 150.


concerning an approximation of the hartree–fock potential by a universal potential function

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