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Relation between total energy and sum of orbital energies for neutral atoms
ChemicalPhysics North-Holland
153 (1991) 141-145
Relation between total energy and sum of orbital energies for neutral atoms A. Nagy Institutefor Theoretical Physics, Kossuth Lajos Umversity, Debrecen, H-4010, Hungary
and N.H. March Theoretical Chemistry Department, University of Oxford, 5 South Parks Road, Oxford OXI 3UB. UK
Received 29 October 1990
The simplest density functional theory, namely the Thomas-Fermi method, leads to the relation E=;E,, where E is the total energy of the neutral atom considered while E, is the sum of the orbital energies. Defining A=E-SE., the quantity A/E versus atomic number Z is studied: (i) the quantity A/E of neutral atoms does not tend to zero with Z in the Hat-tree-Fock theory as one might expect on the grounds of the Thomas-Fermi method, (ii) A/E is not a small quantity in one particular local density approximation ( XLYmethod). Then the value A/E is studied by appropriate use of the 1/Z expansion. Thus, for small Z, if one employs available numerical coefficients for the first three terms in this expansion, one can understand the increase of A/E to a maximum, which is thereby predicted to occur around Z- 9. However, this treatment rapidly breaks down beyond this maximum and one must sum sub-series of the 1/Z expansion to infinite order. When this is done, the analytic result for Z-roe follows. Thus, the main features of the curve of A/E versus Z established by direct numerical calculation can be readily understood for small and large Z.
1. Introduction In the historical development of atomic and especially molecular theory, the relation between the total energy and the sum of orbital energies over occupied states has played a significant role. While studying corrections to the Milne binding energy formula [ 1 ] for neutral atoms, March and Plaskett [ 21 analyzed the relation between the orbital energy sum Es=
1
er,
(1.1)
OWlPled
where E, are the one-electron energies generated by the self-consistent Thomas-Fermi potential energy V(r), and the total energy E. Their result was E=;Es .
(1.2)
The total energy has been often approximated 0301-0104/91/$03.50
by
the aid of the orbital energy sum, especially in semiempirical methods. So the relation between the total energy and the orbital energy sum is of considerable chemical interest. The present work is concerned with a discussion of the origin of the correction A to eq. ( 1.2). Specititally, the quantity A is defined by A=E-$E,,
(1.3)
and following the numerical studies of Kemister and March [ 31 the interest here is to analyse further the behaviour of A/E versus atomic number 2. In particular, one wishes to study the quantity A/E in the Hartree-Fock method as well as in the density functional theory [ 4,5 1. In the latter case numerical results are presented for a particular density functional (Xcu) method. The principal features of the curve A/E for light atoms revealed by numerical calculation, can be nicely
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
A. Nagy, N.H. March / Relatron between total energy and sum of orbital energres
142
accounted for by appropriate sion, pioneered by Hylleraas fruition in the work of Layzer the 1/Z expansion is utilized ber Z in a way in which it is infinite order.
use of the 1/Z expan[ 61 and brought to full and others [ 7,8 1. Then for large atomic numpartially summed up to
2. A/E in the Hartree-Fock method The Hat-tree-Fock
total energy of an atom is given
by Em = i’-+ U,,, + U,, >
(2.1)
where T, U,,, and U,, are the kinetic, nuclear-electronic and electron-electron (including exchange) energies, respectively. The orbital energy sum can be expressed as Es=
C
e,=T+U,,,+2U,,
.
(2.2)
occupxd
Using the virial theorem T=-E,
(2.3)
and eqs. (2.1) and (2.2) we get = -;E+&
ACE-;E,
.
,zg az,'
A -= E
(2.7),wehavefromeq.
-;+;p.
(2.5) (2.8)
In the Thomas-Fermi model [ 111 j?=$ and consequently A= 0. However, in the Hat-tree-Fock theory j3is somewhat larger than 3 (p=2.38 kO.04 [ 10,121). If the total energy E was strictly proportional to a power of Z as shown in eq. (2.6), A/E would have the same constant value for all Z. As, however, eq. (2.6) is not strictly valid, A/E is a function of 2. Indeed, the quantity A/E measures any deviation from the homogeneity property of the total energy. Numerical values of A/E for light atoms have been presented by Kemister and March [ 3 1. Results for the ratio U,,/E have been published also by Fliszar [131. Here the values of A/E for medium and large Z are also presented (fig. 1). Hat-tree-Fock energies are extracted from the work of Froese-Fischer [ 141. Fig. 1 shows that A/E does not go to zero even for large values of Z as one might expect on the grounds of the Thomas-Fermi model. Instead, A/E (and hence p) oscillates around a “constant” value. This “constant” value is somewhat different for intermediate and large values of Z.
(2.4)
The Hellmann-Feynman theorem allows the calculation of the electron-nuclear energy U,, from
une
offormula
(2.4)
0.14
0 12
3
1
+ +
so we arrive at the result A -=E
;+xg
.
(2.5)
N
It has been shown by Gaspar [ 21 that the total Hartree-Fock energy of neutral atoms in their ground state is E=qZB,
(2.6)
with jI= 2.4 and rl is nearly independent of the atomic number for intermediate and large values of Z. Eqs. (2.4) and (2.6) lead to U,,=BE
>
(2.7)
a relation given earlier by Fraga [ IO]. With the help
Fig. 1. Hartree-Fock values for A/E defined by eq. ( 1.3) plotted against the atomic number Z.
143
A. Nagy, N.H. March /Relation between total energy and sum of orbital energies
It is worth mentioning that -tIE aZ,
= ;U.,=v(o),
(2.9)
where V(0) is the electrostatic potential at the nucleus. Using eqs. (2.6) and (2.9) we arrive at the expression E=kZ
(2.10)
V(O),
where k= /3-l. This formula and its generalization for molecules have been given by Politzer [ 15 1. He has also calculated the value of k for several atoms using Hartree-Fock energies [ 16 1.
n(r) Vx,(r)dr
T,-2E,,+
- &
1.
(3.6)
As, unfortunately, we do not know the exact form of the functionals E,, and T,. Presently we cannot calculate precisely the contribution from the third term of eq. (3.6) to A/E. We can, certainly, attempt to make estimates. For instance, we can easily give A/E in the local density approximation. In the Xo method, for example, (3.7)
3. A/E in the density functional theory where The total energy of an atom in the density functional theory [ 1,5 ] can be written as E=Ts+U,,,+Uco+E,,,
(3.1)
where T, is the kinetic energy of the non-interacting electrons, U,,, is the nuclear-electronic potential energy, Uc, is the classical Coulomb potential energy and E,, is the exchange-correlation energy (including correlation kinetic energy). For the orbital energy sum we get Es=Ts+Un,+2Uc,+
n413dr,
is the Xa exchange energy in atomic units [ 181. With the aid of the Hellmann-Feynman theorem [ 181 we obtain the result .
A/E= -f +; /3+EJE
(3.9)
Numerical calculations immediately show that EJE cannot be neglected in the expression of A/E. We have
(3.2) 014
where n(r) is the electronic density and V,, is the exchange-correlation potential defined by
+ 012
0 10
Vxc(r)= v.
(3.8)
(3.3) DOB
I
+ +
+
++ +
+ +
++++ +
The virial theorem [ 121 has the form -“,
E=-T=-T,-T,,
(3.4)
where T, is the difference between the kinetic energy of the interacting and non-interacting electrons. From eqs. (3.1-3.4) we get A=-$+fU,,,-t
[
T,-2E,,+
I
n(r)
Applying again the Hellmann-Feynman find the result
Vxc(r) dr
1.
006
-
+
P
ow-
+
++++
+
+
++
+’ ++ +
+ +
-002
+
(3.5)
theorem we
Fig. 2. The Xa values for (A -I?.) /E plotted against the atomic number Z.
144
‘4. Nagy, N.H March /Relation between total mergv and sum of orbltal energies
calculated A/E for atomic numbers from 3 to 36 using the so-called self-consistent parameter cyscF [ 191. (The values of oscF are very close to the values of aHF [20]. ) Fig. 2 shows the values (A-E,)/E= 5 +2 /3 which can be directly compared with the Hartree-Fock results. The slight difference between the Hartree-Fock and XLYcurves comes from the fact that there is some difference between the Hartree-Fock and the Xa orbitals.
4. Calculation of A from l/Z expansion In this section, we show how the deviation A in eq. ( 1.3 ) can in fact be constructed from the 1/Z expansion. This expansion reads
E(ZN)=ZZ
co(N)+
$,(N)+&(N)+...1 , (4.1)
where the coefficient co(N) is determined completely by the bare Coulomb field solution, while (IV) and t2 (N) are by now known to good accuracy [21]forNintherange2tol5.Ineq.(4.1)wehave treated the case of an atomic ion with atomic number Z and number of electrons N, for reasons that will become clear below, even though the results we are interested in are for neutral atoms with N= Z. Now we note that the coefftcients t, and t2 reflect the self-consistent field, plus the corrections for electron exchange and correlation. These latter effects being relatively small, we shall now employ what is essentially a Hartree argument to enable A to be calculated from eq. (4.1). To gain some understanding of the features of A/E versus Z for small, we shall merely insert the truncated l/Z expansion into eq. (2.5) and after completing the partial differentiation with respect to Z, go on to the neutral atom limit by putting N= Z. Then we find, including coefficients only up to t2 (N):
t,
A -=E
1,: 2
2&(Z) +e, (Z) 2 [ zt,(z)+E,(Z)+E2(Z)/Z
1 .
(4.2)
In this approximation of eq. (4.2) resulting from such a truncation of the 1/Z expansion, we have inserted the coefficients en from table 1 of ref. [ 2 11. We find results which are in semiquantitative accord with the
Fig. 3. Values for A/E defined by eq. ( 1.3) plotted against the atomic number 2. (+ Hartree-Fock, A eq. (4.2) and x eq. (5.4)).
numerical results of Kemister and March [ 31 in that a steep rise of A/E versus Z is found from small negative values to a value N 0.14 at Z= 10. (See fig. 3 ) . However, it is plain from the above study that eq. (4.2) begins to break down rapidly for Z beyond the value Z= 10.
5. Results for A/E in the limit of large Z March and White [22] have demonstrated and later workers have confirmed [ 231, that as the number of electrons N gets large, the coefficients E,(N) have the asymptotic behaviour E,(N)-A,N”+“3.
(5.1)
March and Parr [ 241 have, zation of eq. ( 5.1), obtained the l/Z expansion beyond Fermi energy of atomic ions,
by assuming a generalia formal summation of the original Thomasnamely
E(Z,N)=Z7”A(N/Z)+Z2f(N,Z) +Z5’3f(N,
Z)+-.
(5.2)
The first term on the right-hand side is the original Thomas-Fermi energy; the additional terms are corrections to this limiting form.
A. Nagy, N.H. March /Relation between total energy and sum of orbital energies
Inserting eq. (5.2) into eq. (2.5) yields the result, when we utilize [ 24 ] the values f ; ( 1) = 0, n = 1, 2, 3,
[-;f$(l+f+)] = ^
-I
J 1Iffi;,,3ffi;2,3
.
(5.3)
This yields for neutral atoms of large atomic number Z A -=E
g2z"3+f3
Z”‘fi
+ Z”3& +f3 .
Here, fn denotes f,(N/Z) known: fi=-0.7687,
fi=f,
from the Thomas-Fermi
145
might expect A=0 for large Z. However, A/E calculated by the Hartree-Fock theory is not equal to zero for Z as large as 86. Eq. (5.4) gives an explanation of this behaviour. According this formula A/E decreases very slowly with Z.
Acknowledgement The authors wish to acknowledge the International Center for Theoretical Physics for the support covering their stay in Trieste that made possible this collaboration.
(5.4)
= 1 and these values are
References
(5.5)
[ 1] E.A. Milne, Proc. Cambridge Phil. Sot. 23 (1927) 794. [ 21 N.H. March and J.S. Plaskett, Proc. Roy. Sot. A 234 ( 1956)
theory and its corrections
[ 31 G. Kemister and N.H. March, Chem. Phys. 122 ( 1988) 39.
h=-0.266,
[31. Fig. 3 shows the values of A/E determined by the truncated 1/Z expansion for Z= 1, .... 10 and by eq. (5.4) for Z=41, .... 86. For comparison the HartreeFock values are also shown in fig. 3. The 1/Z expansion predicts for small Z qualitatively correct results for A/E: starting from small and negative and rising rather steeply to a maximum. Eq. (5.4) also leads to a qualitatively correct behaviour.
6. Discussion In conclusion we would like to emphasize that: (i) The quantity A/E of neutral atoms does not fall to zero for large Z in the Hartree-Fock theory. (ii) The quantity A/E of neutral atoms is not small in the local density functional (Xa) theory. This means that the Ruedenberg formula [25,26] (eq. ( 1.2) ) provides a cruder approximation in the Xcu method than in the Hat-tree-Fock theory. (iii) The 1/Z expansion predicts qualitatively correct results for small Z. (iv) The analytical expression ( 5.4) provides results in qualitative agreement with the Hartree-Fock values for large Z. On the grounds of the Thomas-Fermi theory one
419. (41 J.C. Slater, Phys. Rev. 81 (1951) 385. [S] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1135. [6] E.A. Hylleraas, Z. Physik 65 ( 1930) 209. [7] D. Layzer, Ann. Phys. (NY) 8 (1959) 271. [8] R.E. Knight, Phys. Rev. 183 (1969) 45. [ 91 R. Gaspar, Intern. J. Quantum Chem. I ( 1967) 139; ibid., Acta Phys. Chim. De.brecina 16 ( 197 1) 7. [ lo] S. Fraga, Theoret. Chim. Acta 2 ( 1964) 406. [ 1I ] See, e.g., N.H. March, Selfconsistent Fields in Atoms (Pergamon, Oxford, 1975). [ 12 ] R.G. Parr, R.A. Donelly, M. Leng and W.A. Palke, J. Chem. Phys. 68 (1978) 3801. [ 131 S. Flisz&r, Charge distributions and Chemical Effects (Springer, New York, 1983). [ 141 C. Froese-Fischer, At. Data Nucl. DataTables (1972) 301. [ 151 P. Politzer, J. Chem. Phys. 64 (1979) 4239; ibid., 69 (1978) 491. [ 161 P. Politzer, J. Chem. Phys. 70 (1979) 1067. [ 171 M. Levy and J.P. Perdew, Phys. Rev. A 32 (1985) 2010. [ 181 R. Gaspfir and A. Nagy, Intern. J. Quantum Chem. 31 (1987) 639. [19]Seee.g.A.Nagy,Intem.J.QuantumChem.31 (1987)269. [20] K. Schwarz, Phys. Rev. B 5 (1972) 2466. [21] See, e.g., table 1 in the article by N.H. March, Specialist Periodical Reports; Theoretical Chemistry, Vol. 4, Royal Sot. of Chem. (Burlington House, London, 198 1) p. 92. [22] N.H. March and R.J. White, J. Phys. B 5 (1972) 466. [23] Y. Tal and M. Levy, J. Chem. Phys. 72 (1980) 4009. [ 241 See+e.g., the use made by N.H. March and R.G. Parr, Proc. Natl. Acad. Sci. USA 77 (1980) 6285. [25] K. Ruedenberg, J. Chem. Phys. 66 (1977) 375. [26] N.H. March, J. Chem. Phys. 67 (1977) 4618.
153 (1991) 141-145
Relation between total energy and sum of orbital energies for neutral atoms A. Nagy Institutefor Theoretical Physics, Kossuth Lajos Umversity, Debrecen, H-4010, Hungary
and N.H. March Theoretical Chemistry Department, University of Oxford, 5 South Parks Road, Oxford OXI 3UB. UK
Received 29 October 1990
The simplest density functional theory, namely the Thomas-Fermi method, leads to the relation E=;E,, where E is the total energy of the neutral atom considered while E, is the sum of the orbital energies. Defining A=E-SE., the quantity A/E versus atomic number Z is studied: (i) the quantity A/E of neutral atoms does not tend to zero with Z in the Hat-tree-Fock theory as one might expect on the grounds of the Thomas-Fermi method, (ii) A/E is not a small quantity in one particular local density approximation ( XLYmethod). Then the value A/E is studied by appropriate use of the 1/Z expansion. Thus, for small Z, if one employs available numerical coefficients for the first three terms in this expansion, one can understand the increase of A/E to a maximum, which is thereby predicted to occur around Z- 9. However, this treatment rapidly breaks down beyond this maximum and one must sum sub-series of the 1/Z expansion to infinite order. When this is done, the analytic result for Z-roe follows. Thus, the main features of the curve of A/E versus Z established by direct numerical calculation can be readily understood for small and large Z.
1. Introduction In the historical development of atomic and especially molecular theory, the relation between the total energy and the sum of orbital energies over occupied states has played a significant role. While studying corrections to the Milne binding energy formula [ 1 ] for neutral atoms, March and Plaskett [ 21 analyzed the relation between the orbital energy sum Es=
1
er,
(1.1)
OWlPled
where E, are the one-electron energies generated by the self-consistent Thomas-Fermi potential energy V(r), and the total energy E. Their result was E=;Es .
(1.2)
The total energy has been often approximated 0301-0104/91/$03.50
by
the aid of the orbital energy sum, especially in semiempirical methods. So the relation between the total energy and the orbital energy sum is of considerable chemical interest. The present work is concerned with a discussion of the origin of the correction A to eq. ( 1.2). Specititally, the quantity A is defined by A=E-$E,,
(1.3)
and following the numerical studies of Kemister and March [ 31 the interest here is to analyse further the behaviour of A/E versus atomic number 2. In particular, one wishes to study the quantity A/E in the Hartree-Fock method as well as in the density functional theory [ 4,5 1. In the latter case numerical results are presented for a particular density functional (Xcu) method. The principal features of the curve A/E for light atoms revealed by numerical calculation, can be nicely
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
A. Nagy, N.H. March / Relatron between total energy and sum of orbital energres
142
accounted for by appropriate sion, pioneered by Hylleraas fruition in the work of Layzer the 1/Z expansion is utilized ber Z in a way in which it is infinite order.
use of the 1/Z expan[ 61 and brought to full and others [ 7,8 1. Then for large atomic numpartially summed up to
2. A/E in the Hartree-Fock method The Hat-tree-Fock
total energy of an atom is given
by Em = i’-+ U,,, + U,, >
(2.1)
where T, U,,, and U,, are the kinetic, nuclear-electronic and electron-electron (including exchange) energies, respectively. The orbital energy sum can be expressed as Es=
C
e,=T+U,,,+2U,,
.
(2.2)
occupxd
Using the virial theorem T=-E,
(2.3)
and eqs. (2.1) and (2.2) we get = -;E+&
ACE-;E,
.
,zg az,'
A -= E
(2.7),wehavefromeq.
-;+;p.
(2.5) (2.8)
In the Thomas-Fermi model [ 111 j?=$ and consequently A= 0. However, in the Hat-tree-Fock theory j3is somewhat larger than 3 (p=2.38 kO.04 [ 10,121). If the total energy E was strictly proportional to a power of Z as shown in eq. (2.6), A/E would have the same constant value for all Z. As, however, eq. (2.6) is not strictly valid, A/E is a function of 2. Indeed, the quantity A/E measures any deviation from the homogeneity property of the total energy. Numerical values of A/E for light atoms have been presented by Kemister and March [ 3 1. Results for the ratio U,,/E have been published also by Fliszar [131. Here the values of A/E for medium and large Z are also presented (fig. 1). Hat-tree-Fock energies are extracted from the work of Froese-Fischer [ 141. Fig. 1 shows that A/E does not go to zero even for large values of Z as one might expect on the grounds of the Thomas-Fermi model. Instead, A/E (and hence p) oscillates around a “constant” value. This “constant” value is somewhat different for intermediate and large values of Z.
(2.4)
The Hellmann-Feynman theorem allows the calculation of the electron-nuclear energy U,, from
une
offormula
(2.4)
0.14
0 12
3
1
+ +
so we arrive at the result A -=E
;+xg
.
(2.5)
N
It has been shown by Gaspar [ 21 that the total Hartree-Fock energy of neutral atoms in their ground state is E=qZB,
(2.6)
with jI= 2.4 and rl is nearly independent of the atomic number for intermediate and large values of Z. Eqs. (2.4) and (2.6) lead to U,,=BE
>
(2.7)
a relation given earlier by Fraga [ IO]. With the help
Fig. 1. Hartree-Fock values for A/E defined by eq. ( 1.3) plotted against the atomic number Z.
143
A. Nagy, N.H. March /Relation between total energy and sum of orbital energies
It is worth mentioning that -tIE aZ,
= ;U.,=v(o),
(2.9)
where V(0) is the electrostatic potential at the nucleus. Using eqs. (2.6) and (2.9) we arrive at the expression E=kZ
(2.10)
V(O),
where k= /3-l. This formula and its generalization for molecules have been given by Politzer [ 15 1. He has also calculated the value of k for several atoms using Hartree-Fock energies [ 16 1.
n(r) Vx,(r)dr
T,-2E,,+
- &
1.
(3.6)
As, unfortunately, we do not know the exact form of the functionals E,, and T,. Presently we cannot calculate precisely the contribution from the third term of eq. (3.6) to A/E. We can, certainly, attempt to make estimates. For instance, we can easily give A/E in the local density approximation. In the Xo method, for example, (3.7)
3. A/E in the density functional theory where The total energy of an atom in the density functional theory [ 1,5 ] can be written as E=Ts+U,,,+Uco+E,,,
(3.1)
where T, is the kinetic energy of the non-interacting electrons, U,,, is the nuclear-electronic potential energy, Uc, is the classical Coulomb potential energy and E,, is the exchange-correlation energy (including correlation kinetic energy). For the orbital energy sum we get Es=Ts+Un,+2Uc,+
n413dr,
is the Xa exchange energy in atomic units [ 181. With the aid of the Hellmann-Feynman theorem [ 181 we obtain the result .
A/E= -f +; /3+EJE
(3.9)
Numerical calculations immediately show that EJE cannot be neglected in the expression of A/E. We have
(3.2) 014
where n(r) is the electronic density and V,, is the exchange-correlation potential defined by
+ 012
0 10
Vxc(r)= v.
(3.8)
(3.3) DOB
I
+ +
+
++ +
+ +
++++ +
The virial theorem [ 121 has the form -“,
E=-T=-T,-T,,
(3.4)
where T, is the difference between the kinetic energy of the interacting and non-interacting electrons. From eqs. (3.1-3.4) we get A=-$+fU,,,-t
[
T,-2E,,+
I
n(r)
Applying again the Hellmann-Feynman find the result
Vxc(r) dr
1.
006
-
+
P
ow-
+
++++
+
+
++
+’ ++ +
+ +
-002
+
(3.5)
theorem we
Fig. 2. The Xa values for (A -I?.) /E plotted against the atomic number Z.
144
‘4. Nagy, N.H March /Relation between total mergv and sum of orbltal energies
calculated A/E for atomic numbers from 3 to 36 using the so-called self-consistent parameter cyscF [ 191. (The values of oscF are very close to the values of aHF [20]. ) Fig. 2 shows the values (A-E,)/E= 5 +2 /3 which can be directly compared with the Hartree-Fock results. The slight difference between the Hartree-Fock and XLYcurves comes from the fact that there is some difference between the Hartree-Fock and the Xa orbitals.
4. Calculation of A from l/Z expansion In this section, we show how the deviation A in eq. ( 1.3 ) can in fact be constructed from the 1/Z expansion. This expansion reads
E(ZN)=ZZ
co(N)+
$,(N)+&(N)+...1 , (4.1)
where the coefficient co(N) is determined completely by the bare Coulomb field solution, while (IV) and t2 (N) are by now known to good accuracy [21]forNintherange2tol5.Ineq.(4.1)wehave treated the case of an atomic ion with atomic number Z and number of electrons N, for reasons that will become clear below, even though the results we are interested in are for neutral atoms with N= Z. Now we note that the coefftcients t, and t2 reflect the self-consistent field, plus the corrections for electron exchange and correlation. These latter effects being relatively small, we shall now employ what is essentially a Hartree argument to enable A to be calculated from eq. (4.1). To gain some understanding of the features of A/E versus Z for small, we shall merely insert the truncated l/Z expansion into eq. (2.5) and after completing the partial differentiation with respect to Z, go on to the neutral atom limit by putting N= Z. Then we find, including coefficients only up to t2 (N):
t,
A -=E
1,: 2
2&(Z) +e, (Z) 2 [ zt,(z)+E,(Z)+E2(Z)/Z
1 .
(4.2)
In this approximation of eq. (4.2) resulting from such a truncation of the 1/Z expansion, we have inserted the coefficients en from table 1 of ref. [ 2 11. We find results which are in semiquantitative accord with the
Fig. 3. Values for A/E defined by eq. ( 1.3) plotted against the atomic number 2. (+ Hartree-Fock, A eq. (4.2) and x eq. (5.4)).
numerical results of Kemister and March [ 31 in that a steep rise of A/E versus Z is found from small negative values to a value N 0.14 at Z= 10. (See fig. 3 ) . However, it is plain from the above study that eq. (4.2) begins to break down rapidly for Z beyond the value Z= 10.
5. Results for A/E in the limit of large Z March and White [22] have demonstrated and later workers have confirmed [ 231, that as the number of electrons N gets large, the coefficients E,(N) have the asymptotic behaviour E,(N)-A,N”+“3.
(5.1)
March and Parr [ 241 have, zation of eq. ( 5.1), obtained the l/Z expansion beyond Fermi energy of atomic ions,
by assuming a generalia formal summation of the original Thomasnamely
E(Z,N)=Z7”A(N/Z)+Z2f(N,Z) +Z5’3f(N,
Z)+-.
(5.2)
The first term on the right-hand side is the original Thomas-Fermi energy; the additional terms are corrections to this limiting form.
A. Nagy, N.H. March /Relation between total energy and sum of orbital energies
Inserting eq. (5.2) into eq. (2.5) yields the result, when we utilize [ 24 ] the values f ; ( 1) = 0, n = 1, 2, 3,
[-;f$(l+f+)] = ^
-I
J 1Iffi;,,3ffi;2,3
.
(5.3)
This yields for neutral atoms of large atomic number Z A -=E
g2z"3+f3
Z”‘fi
+ Z”3& +f3 .
Here, fn denotes f,(N/Z) known: fi=-0.7687,
fi=f,
from the Thomas-Fermi
145
might expect A=0 for large Z. However, A/E calculated by the Hartree-Fock theory is not equal to zero for Z as large as 86. Eq. (5.4) gives an explanation of this behaviour. According this formula A/E decreases very slowly with Z.
Acknowledgement The authors wish to acknowledge the International Center for Theoretical Physics for the support covering their stay in Trieste that made possible this collaboration.
(5.4)
= 1 and these values are
References
(5.5)
[ 1] E.A. Milne, Proc. Cambridge Phil. Sot. 23 (1927) 794. [ 21 N.H. March and J.S. Plaskett, Proc. Roy. Sot. A 234 ( 1956)
theory and its corrections
[ 31 G. Kemister and N.H. March, Chem. Phys. 122 ( 1988) 39.
h=-0.266,
[31. Fig. 3 shows the values of A/E determined by the truncated 1/Z expansion for Z= 1, .... 10 and by eq. (5.4) for Z=41, .... 86. For comparison the HartreeFock values are also shown in fig. 3. The 1/Z expansion predicts for small Z qualitatively correct results for A/E: starting from small and negative and rising rather steeply to a maximum. Eq. (5.4) also leads to a qualitatively correct behaviour.
6. Discussion In conclusion we would like to emphasize that: (i) The quantity A/E of neutral atoms does not fall to zero for large Z in the Hartree-Fock theory. (ii) The quantity A/E of neutral atoms is not small in the local density functional (Xa) theory. This means that the Ruedenberg formula [25,26] (eq. ( 1.2) ) provides a cruder approximation in the Xcu method than in the Hat-tree-Fock theory. (iii) The 1/Z expansion predicts qualitatively correct results for small Z. (iv) The analytical expression ( 5.4) provides results in qualitative agreement with the Hartree-Fock values for large Z. On the grounds of the Thomas-Fermi theory one
419. (41 J.C. Slater, Phys. Rev. 81 (1951) 385. [S] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1135. [6] E.A. Hylleraas, Z. Physik 65 ( 1930) 209. [7] D. Layzer, Ann. Phys. (NY) 8 (1959) 271. [8] R.E. Knight, Phys. Rev. 183 (1969) 45. [ 91 R. Gaspar, Intern. J. Quantum Chem. I ( 1967) 139; ibid., Acta Phys. Chim. De.brecina 16 ( 197 1) 7. [ lo] S. Fraga, Theoret. Chim. Acta 2 ( 1964) 406. [ 1I ] See, e.g., N.H. March, Selfconsistent Fields in Atoms (Pergamon, Oxford, 1975). [ 12 ] R.G. Parr, R.A. Donelly, M. Leng and W.A. Palke, J. Chem. Phys. 68 (1978) 3801. [ 131 S. Flisz&r, Charge distributions and Chemical Effects (Springer, New York, 1983). [ 141 C. Froese-Fischer, At. Data Nucl. DataTables (1972) 301. [ 151 P. Politzer, J. Chem. Phys. 64 (1979) 4239; ibid., 69 (1978) 491. [ 161 P. Politzer, J. Chem. Phys. 70 (1979) 1067. [ 171 M. Levy and J.P. Perdew, Phys. Rev. A 32 (1985) 2010. [ 181 R. Gaspfir and A. Nagy, Intern. J. Quantum Chem. 31 (1987) 639. [19]Seee.g.A.Nagy,Intem.J.QuantumChem.31 (1987)269. [20] K. Schwarz, Phys. Rev. B 5 (1972) 2466. [21] See, e.g., table 1 in the article by N.H. March, Specialist Periodical Reports; Theoretical Chemistry, Vol. 4, Royal Sot. of Chem. (Burlington House, London, 198 1) p. 92. [22] N.H. March and R.J. White, J. Phys. B 5 (1972) 466. [23] Y. Tal and M. Levy, J. Chem. Phys. 72 (1980) 4009. [ 241 See+e.g., the use made by N.H. March and R.G. Parr, Proc. Natl. Acad. Sci. USA 77 (1980) 6285. [25] K. Ruedenberg, J. Chem. Phys. 66 (1977) 375. [26] N.H. March, J. Chem. Phys. 67 (1977) 4618.