- Email: [email protected]
Extended Thomas-Fermi approach to diatomic systems
Volume hA, number 1
PHYSICS LETTERS
16 April 1979
EXTENDED THOMAS—FERMI APPROACH TO DIATOMIC SYSTEMS ~ R.M. DREIZLER, E.K.U. GROSS and A. TOEPFER Institut für Theoretische Physik der Universitat, Frankfurt/Main, West Germany Received 3 October 1978
An accurate, analytical approximation to the solution of the TFDW equation for two-centre boundary conditions is presented. Correlation diagrams can then be calculated with an effective single-particle potential derived from the TFDW density. Total energies and orbital energies for the N—N system are compared with corresponding HF results.
Quantum statistical approaches to the many-body problem in nuclear, atomic and molecular structure physics are distinguished by an extensive history. The detailed consideration of the socalled quantum correction first discussed by von Weizsäcker [1] and reformulated by Kirzhnits [2] is, however, less extensive. The only results presented so far for atomic and quasimolecular diatomic systems are due to Yonei et al. [3]. We investigate the Thomas—Fermi—Dirac--von Weizsäcker (TFDW) approach in its application to the N—N system, the aim being (1) the calculation of the total electronic energy as a function of the internuclear separation, and (2) the calculation of correlation diagrams based on an effective single-particle potential extracted from the quantum statistical density (an approach initiated on a more phenomenological level by Eichler and Wille [4]). In the TFDW energy functional (valid for both atoms and quasimolecules), ETFDW = fdr(cip5/3 —c 2p’ +pW 2/p) (1) + ~PVe+ c3(Vp) with the nuclear potential W, the electronic potential ,
,
~.
Ve(r)
_,
~\
p.~r,
3,
i Ir r’I d r —
Supported
in part by
1I3, =
j~(3i~2)2f3
c2
=
~(3/ir)
c 3
=
X18
(all quantitiesthroughout this paper are given in the atomic unit system), we adopt Yonei’s choice of A = 1/5, with the interpretation, that it provides a means of adjustment for the higher-order terms of the gradient expansion. Variation of the energy functional ETFDW with respect to the density p and the subsidiary condition of fixed particle number yields the equation ~ 2/3 ~ 1/3 ÷w + v + v ~C1 P ~C2P e 0 —
+ c3
((Vp/p)
2 —
2Vp/p) = 0
,
(2)
with V0 being a Lagrange parameter. It can be shown [3] that the asymptotic behaviour of any solution p of this equation for a neutral system is given by 2exp[—(V 2r]. (3) p r 0/c3)~’ charge Further, centres, the density whichcan is abe definite assumed improvement to be finite comat the pared with the standard TF-density. The total electrostatic potential V of the TFDW-. atom is most practically discussed in terms of a screening function ~ defined by .
,
and *
c1
.
V = ZØ(r)/r, and the boundary condition
(4)
the Deutsche Forschungsgemeinschaft.
49
Volume 71A, number 1
0(0)
=
1
PHYSICS LETTERS
(5)
.
16 April 1979
-~
With the aid of Poisson’s equation we can establish a connection between the density p and the screening function ~ by p=~ Ve=~(V
W)~-~(1—0(r))
200 -240
1 Z~”(r) = ~ r for r ~ 0. (6) We found that a fast and accurate solution of eq. (2) can be obtained in terms of an analytical approximation to the screening function A suitably constructed analytical form for depending on a set of parameters, is inserted into eq. (6). The energy functional (1) is then minimized as a function of the parameters. For the analytical approximation of ~ we have chosen the following form:
________
—H~-~---±—±~--~ I
a
~.
0
~,
05 10
(7) 1 + br ÷ cr2 + dr2e°’~ It fulfills the required boundary conditions (3) and (5) and provides a finite density at the origin, if the relation d = b (b a) c is satisfied by the parameters of eq. (7). As an illustration we present in table I the parameters and the total energy for the nitrogen atom. We find good agreement with the corresponding HF-result [5]. For optimization of the parameters we use an —
04
00
2
6
I
20
74
6--.Fig. 1. The total energy of the TFDW-quasimolecule N—N in comparison to HF-results (dashed curve). The lower part of the graph gives the percentage deviation (100 (ETFDW — EHF)/ I EJ~~ I).
—
mized for each internuclear distance R separately. Via the ansatz (8) rotational symmetry with respect to the internuclear axis, the presence of the pure Coulomb potential at both centres as well as the normalisation condition f pdr = +Z 2 are guaranteed. Fig. 1 shows the total electronic energies for the N—N system in comparison with HF-results. Again the agreement is excellent (better than 1%, except for the value at R = 0.1). The contribution of the von Weizsdcker term to the total energy amounts to 10—14%. The TFDW model so far provides a means for the calculation of an approximate total electron density or, equivalently, the total electrostatic potential V of the system. For the calculation of correlation diagrams the two-centre Schrodinger equation is to be solved with an effective single-particle potential Veff. In a previous publication [7] we proposed the two-centre TF poten-
evolutionary strategy, a very effective and fast procedure, proposed by Rechenberg [6]. In order to obtain an analytical approximation in the case ofdiatomic systems for the two-centre TFDWpotential, we superimpose two screened Coulomb potentials, V 0~(r~)Z1/r1 + 02(r2)Z2/r2, (8) where the analytical form of the individual screening functions 01 and 02 is still given by eq. (7). However, the parameters in Ø~and 02 are not to be taken from the atomic case, instead, the parameters must be optiTable 1 Parameters and total energy for the nitrogen atom.
50
-~
0
—
a
a
b
1.334
45.991
48~57
-—
—-
—
-
--
94.190
—54.862
—54.401
-
Volume 71A, number 1
PHYSICS LETTERS
tial (which is also an approximation to the total electrostatic potential) as a possible effective potential. For further improvements it is, however, necessary to consider modifications. We adopt the point of view, that the Hartree—Fock— Slater approximation provides an adequate prescription for the choice of an effective single-particle potential, where in place of the self-consistent density the density obtained in the TFDW approximation can be inserted. In addition to the direct term 113 we thus and,consider as used the Slater exchange term [8] ~((3/ir)p) by several authors [91a self-energy correction for the asymptotic region. These corrections are taken into account by the following definition of the effective potential: Z 101(r1) Z2Ø2(r2) Veff= r1 r2
16 April 1979 -
Ii
-
3~
-1
liT — 0
,
-in
1
1r~
—
3 ~3 (o~(r1)Z1 2 4~.2 r1 _________
ifVeff
Veff =
(1 >max—
max
,
1)
+
_________
05
10
15
20
R Lou.] Fig. 2. Correlation diagram for the N—N system calculated on
1
the basis of the TFDW-model.
dence of this parameter by minimising the groundstate
if Veff
We point out that the effective single -particle potential (9) (in contrast to the two-centre TFDW-potential (8)) cannot be split into two spherically symmetric parts. The corresponding two-centre Schrodinger equation, heff(~Ol(r,R)
-100 Il~
0~r2)Z2)1”3
e~(R)p~(r,R), (10) 2 + Veff, is solved by diagonalization, with ~V whereheff the=eigenfunctions ‘p 1 are expanded in terms of the basis set iJinIm first introduced by Hylleraas [10] : 2e(_I_l)/a ~ ~) (~2 l)m/ X L~°((~1)/a)Pr(7?)e’m~’. (11) i~and ‘p are the well-known prolate spheroidal coordinates, the functions L~mare the generalised Laguerre polynomials, and the P7°the associated Legendre functions. The parameter a provides a means of adjustment of the basis functions to the particular internuclear separation R. We optimize the R-depen-
energy ~k(O~)ek(R) as a function of a for each internuclear distance R separately. The matrix elements of the kinetic energy and, as the basis is nonorthogonal, of the overlap can be calculated analytically and were first given by Eichler and Wille [4] The matrix elements of the effective potential are evaluated numerically by a Gauss—Kronrod procedure. In fig. 2 we show the correlation diagram for the N—N system. A more detailed comparison with cor.
Table 2
—
Comparison of electronic orbital energies for the N—N system al
R = 0.6 au. The orbital energies CTFDW are calculated as solution [101. ref. of eq. (10), the Hartree—Fock values EHF are taken from
—
6HF
—
lOg llTu
2Og 2~u 30g
20.47 16.24 2.58 2d3 0.97 0.20
~TFDW 19.50 15.50 2.23 1.65
0.67 0.16
51
Volume 71A, number 1
PHYSICS LETTERS
responding HF-results [10]is presented in table 2, where the values of the orbital energies are listed for the particular internuclear separation R = 0.6 au. The comparison is very favourable. In fig. 3 we compare the TFDW-density for the same
-
separation (R = 0.6 au) with the quantum mechanical density p = ~ calculated with the solution of eq. (10). There is reasonable agreement. A slightly different structure in the form of stronger necking, which is due to the effect of the ungeradeorbital contribution, is found in the quantum mechanical density. As a check of the consistency of the calculation we reevaluate the total energy
--
E(R)=(~lHII),
T
4 -
-
-
I
(12) =
det (‘pt) con-
structed from the solution of the eigenvalue problem (10). H is the “exact” electronic hamiltonian
-
I
-2
internuclear
with a determinantal wave function c1
-
-
I
16 April 1979
I
H=
-
-
~-~V? 2m
--
--
1
~
/
--
1r1+RI
~ /
z2__
r1—RI
(13) -8
~- --~----~
6
4
2
0
~~----~----~
2
4
8
6
(a) 80
~
so
I
I
-
40
-40 -60 -8C
-
-
-so
-
.1
-60
-40
-20
0
.1
2.0
40
60
00 *E-1
(13)
Fig. 3. Density contour plot for N—N at the internuclear separation R = 0.6 au. (a) TFDW-density. (b) Quantum mechanical density. The contour lines correspond top = 5,6,7,9,11,15, 25,50,100 au.
52
The value compares of the total energy for Rwith = 0.6 au, E(R) = —170.47, satisfactorily ETFDW = —173.05 and the corresponding HF-energy EIIF = --173.42. Systematic investigation of heavier and of heteronuclear systems is envisaged. A relativistic extension based on the Dirac equation is under consideration. It should prove to be valuable for spectroscopic aspects of heavy-ion collisions.
[1] C.F. von Weizalcker, Z. Phys. 96 (1935) 431. [2] D.A. Kirzhnits, Field theoretical methods in many-body systems (Pergamon, Oxford, 1967). [31K. Yonej andY. Tomishima, J. Phys. Soc. Japan 20 (1965) 1051; Y. Tomishima and K. Yonei, J. Phys. Soc. Japan 21 (1966)142; K. Yonei, J. Phys. Soc. Japan 22 (1967) 1127; 31(1971) 882. [4] J. Eichler and U. Wile, Phys. Rev. Lett. 33 (1974) 56; Phys. Rev. All (1975) 1973. [5] E. Clementi and C. Roetti, At. Data Nuci. Data Tables 14 (1974) 177.
Volume hA, number 1
PHYSICS LETTERS
[6] I. Rechenberg, Evolutionsstrategie, Problemata 15 (Frommann-Holzboog, Stuttgart, 1973). [7] E. Gross and R.M. Dreizler, Phys. Lett. 57A (1976) 131. [8] J.C. Slater, Phys. Rev. 81(1951) 385. [9] R. Latter, Phys. Rev. 99 (1955) 510; F. Hermann and S. Skullman, Atomic structure calculations (Prentice Hall, Englewood Cliffs, NJ, 1963).
16 April 1979
[10] E. Hylleraas, Z. Phys. 71(1931) 739. [11] W.C. Ermler and R.S. Mulliken, J. Chem. Phys. 66 (1977) 3031. [12] V.K. Nikulin and N.A. Gruschina, J. Phys. B11 (1978) 3553.
53_
PHYSICS LETTERS
16 April 1979
EXTENDED THOMAS—FERMI APPROACH TO DIATOMIC SYSTEMS ~ R.M. DREIZLER, E.K.U. GROSS and A. TOEPFER Institut für Theoretische Physik der Universitat, Frankfurt/Main, West Germany Received 3 October 1978
An accurate, analytical approximation to the solution of the TFDW equation for two-centre boundary conditions is presented. Correlation diagrams can then be calculated with an effective single-particle potential derived from the TFDW density. Total energies and orbital energies for the N—N system are compared with corresponding HF results.
Quantum statistical approaches to the many-body problem in nuclear, atomic and molecular structure physics are distinguished by an extensive history. The detailed consideration of the socalled quantum correction first discussed by von Weizsäcker [1] and reformulated by Kirzhnits [2] is, however, less extensive. The only results presented so far for atomic and quasimolecular diatomic systems are due to Yonei et al. [3]. We investigate the Thomas—Fermi—Dirac--von Weizsäcker (TFDW) approach in its application to the N—N system, the aim being (1) the calculation of the total electronic energy as a function of the internuclear separation, and (2) the calculation of correlation diagrams based on an effective single-particle potential extracted from the quantum statistical density (an approach initiated on a more phenomenological level by Eichler and Wille [4]). In the TFDW energy functional (valid for both atoms and quasimolecules), ETFDW = fdr(cip5/3 —c 2p’ +pW 2/p) (1) + ~PVe+ c3(Vp) with the nuclear potential W, the electronic potential ,
,
~.
Ve(r)
_,
~\
p.~r,
3,
i Ir r’I d r —
Supported
in part by
1I3, =
j~(3i~2)2f3
c2
=
~(3/ir)
c 3
=
X18
(all quantitiesthroughout this paper are given in the atomic unit system), we adopt Yonei’s choice of A = 1/5, with the interpretation, that it provides a means of adjustment for the higher-order terms of the gradient expansion. Variation of the energy functional ETFDW with respect to the density p and the subsidiary condition of fixed particle number yields the equation ~ 2/3 ~ 1/3 ÷w + v + v ~C1 P ~C2P e 0 —
+ c3
((Vp/p)
2 —
2Vp/p) = 0
,
(2)
with V0 being a Lagrange parameter. It can be shown [3] that the asymptotic behaviour of any solution p of this equation for a neutral system is given by 2exp[—(V 2r]. (3) p r 0/c3)~’ charge Further, centres, the density whichcan is abe definite assumed improvement to be finite comat the pared with the standard TF-density. The total electrostatic potential V of the TFDW-. atom is most practically discussed in terms of a screening function ~ defined by .
,
and *
c1
.
V = ZØ(r)/r, and the boundary condition
(4)
the Deutsche Forschungsgemeinschaft.
49
Volume 71A, number 1
0(0)
=
1
PHYSICS LETTERS
(5)
.
16 April 1979
-~
With the aid of Poisson’s equation we can establish a connection between the density p and the screening function ~ by p=~ Ve=~(V
W)~-~(1—0(r))
200 -240
1 Z~”(r) = ~ r for r ~ 0. (6) We found that a fast and accurate solution of eq. (2) can be obtained in terms of an analytical approximation to the screening function A suitably constructed analytical form for depending on a set of parameters, is inserted into eq. (6). The energy functional (1) is then minimized as a function of the parameters. For the analytical approximation of ~ we have chosen the following form:
________
—H~-~---±—±~--~ I
a
~.
0
~,
05 10
(7) 1 + br ÷ cr2 + dr2e°’~ It fulfills the required boundary conditions (3) and (5) and provides a finite density at the origin, if the relation d = b (b a) c is satisfied by the parameters of eq. (7). As an illustration we present in table I the parameters and the total energy for the nitrogen atom. We find good agreement with the corresponding HF-result [5]. For optimization of the parameters we use an —
04
00
2
6
I
20
74
6--.Fig. 1. The total energy of the TFDW-quasimolecule N—N in comparison to HF-results (dashed curve). The lower part of the graph gives the percentage deviation (100 (ETFDW — EHF)/ I EJ~~ I).
—
mized for each internuclear distance R separately. Via the ansatz (8) rotational symmetry with respect to the internuclear axis, the presence of the pure Coulomb potential at both centres as well as the normalisation condition f pdr = +Z 2 are guaranteed. Fig. 1 shows the total electronic energies for the N—N system in comparison with HF-results. Again the agreement is excellent (better than 1%, except for the value at R = 0.1). The contribution of the von Weizsdcker term to the total energy amounts to 10—14%. The TFDW model so far provides a means for the calculation of an approximate total electron density or, equivalently, the total electrostatic potential V of the system. For the calculation of correlation diagrams the two-centre Schrodinger equation is to be solved with an effective single-particle potential Veff. In a previous publication [7] we proposed the two-centre TF poten-
evolutionary strategy, a very effective and fast procedure, proposed by Rechenberg [6]. In order to obtain an analytical approximation in the case ofdiatomic systems for the two-centre TFDWpotential, we superimpose two screened Coulomb potentials, V 0~(r~)Z1/r1 + 02(r2)Z2/r2, (8) where the analytical form of the individual screening functions 01 and 02 is still given by eq. (7). However, the parameters in Ø~and 02 are not to be taken from the atomic case, instead, the parameters must be optiTable 1 Parameters and total energy for the nitrogen atom.
50
-~
0
—
a
a
b
1.334
45.991
48~57
-—
—-
—
-
--
94.190
—54.862
—54.401
-
Volume 71A, number 1
PHYSICS LETTERS
tial (which is also an approximation to the total electrostatic potential) as a possible effective potential. For further improvements it is, however, necessary to consider modifications. We adopt the point of view, that the Hartree—Fock— Slater approximation provides an adequate prescription for the choice of an effective single-particle potential, where in place of the self-consistent density the density obtained in the TFDW approximation can be inserted. In addition to the direct term 113 we thus and,consider as used the Slater exchange term [8] ~((3/ir)p) by several authors [91a self-energy correction for the asymptotic region. These corrections are taken into account by the following definition of the effective potential: Z 101(r1) Z2Ø2(r2) Veff= r1 r2
16 April 1979 -
Ii
-
3~
-1
liT — 0
,
-in
1
1r~
—
3 ~3 (o~(r1)Z1 2 4~.2 r1 _________
ifVeff
Veff =
(1 >max—
max
,
1)
+
_________
05
10
15
20
R Lou.] Fig. 2. Correlation diagram for the N—N system calculated on
1
the basis of the TFDW-model.
dence of this parameter by minimising the groundstate
if Veff
We point out that the effective single -particle potential (9) (in contrast to the two-centre TFDW-potential (8)) cannot be split into two spherically symmetric parts. The corresponding two-centre Schrodinger equation, heff(~Ol(r,R)
-100 Il~
0~r2)Z2)1”3
e~(R)p~(r,R), (10) 2 + Veff, is solved by diagonalization, with ~V whereheff the=eigenfunctions ‘p 1 are expanded in terms of the basis set iJinIm first introduced by Hylleraas [10] : 2e(_I_l)/a ~ ~) (~2 l)m/ X L~°((~1)/a)Pr(7?)e’m~’. (11) i~and ‘p are the well-known prolate spheroidal coordinates, the functions L~mare the generalised Laguerre polynomials, and the P7°the associated Legendre functions. The parameter a provides a means of adjustment of the basis functions to the particular internuclear separation R. We optimize the R-depen-
energy ~k(O~)ek(R) as a function of a for each internuclear distance R separately. The matrix elements of the kinetic energy and, as the basis is nonorthogonal, of the overlap can be calculated analytically and were first given by Eichler and Wille [4] The matrix elements of the effective potential are evaluated numerically by a Gauss—Kronrod procedure. In fig. 2 we show the correlation diagram for the N—N system. A more detailed comparison with cor.
Table 2
—
Comparison of electronic orbital energies for the N—N system al
R = 0.6 au. The orbital energies CTFDW are calculated as solution [101. ref. of eq. (10), the Hartree—Fock values EHF are taken from
—
6HF
—
lOg llTu
2Og 2~u 30g
20.47 16.24 2.58 2d3 0.97 0.20
~TFDW 19.50 15.50 2.23 1.65
0.67 0.16
51
Volume 71A, number 1
PHYSICS LETTERS
responding HF-results [10]is presented in table 2, where the values of the orbital energies are listed for the particular internuclear separation R = 0.6 au. The comparison is very favourable. In fig. 3 we compare the TFDW-density for the same
-
separation (R = 0.6 au) with the quantum mechanical density p = ~ calculated with the solution of eq. (10). There is reasonable agreement. A slightly different structure in the form of stronger necking, which is due to the effect of the ungeradeorbital contribution, is found in the quantum mechanical density. As a check of the consistency of the calculation we reevaluate the total energy
--
E(R)=(~lHII),
T
4 -
-
-
I
(12) =
det (‘pt) con-
structed from the solution of the eigenvalue problem (10). H is the “exact” electronic hamiltonian
-
I
-2
internuclear
with a determinantal wave function c1
-
-
I
16 April 1979
I
H=
-
-
~-~V? 2m
--
--
1
~
/
--
1r1+RI
~ /
z2__
r1—RI
(13) -8
~- --~----~
6
4
2
0
~~----~----~
2
4
8
6
(a) 80
~
so
I
I
-
40
-40 -60 -8C
-
-
-so
-
.1
-60
-40
-20
0
.1
2.0
40
60
00 *E-1
(13)
Fig. 3. Density contour plot for N—N at the internuclear separation R = 0.6 au. (a) TFDW-density. (b) Quantum mechanical density. The contour lines correspond top = 5,6,7,9,11,15, 25,50,100 au.
52
The value compares of the total energy for Rwith = 0.6 au, E(R) = —170.47, satisfactorily ETFDW = —173.05 and the corresponding HF-energy EIIF = --173.42. Systematic investigation of heavier and of heteronuclear systems is envisaged. A relativistic extension based on the Dirac equation is under consideration. It should prove to be valuable for spectroscopic aspects of heavy-ion collisions.
[1] C.F. von Weizalcker, Z. Phys. 96 (1935) 431. [2] D.A. Kirzhnits, Field theoretical methods in many-body systems (Pergamon, Oxford, 1967). [31K. Yonej andY. Tomishima, J. Phys. Soc. Japan 20 (1965) 1051; Y. Tomishima and K. Yonei, J. Phys. Soc. Japan 21 (1966)142; K. Yonei, J. Phys. Soc. Japan 22 (1967) 1127; 31(1971) 882. [4] J. Eichler and U. Wile, Phys. Rev. Lett. 33 (1974) 56; Phys. Rev. All (1975) 1973. [5] E. Clementi and C. Roetti, At. Data Nuci. Data Tables 14 (1974) 177.
Volume hA, number 1
PHYSICS LETTERS
[6] I. Rechenberg, Evolutionsstrategie, Problemata 15 (Frommann-Holzboog, Stuttgart, 1973). [7] E. Gross and R.M. Dreizler, Phys. Lett. 57A (1976) 131. [8] J.C. Slater, Phys. Rev. 81(1951) 385. [9] R. Latter, Phys. Rev. 99 (1955) 510; F. Hermann and S. Skullman, Atomic structure calculations (Prentice Hall, Englewood Cliffs, NJ, 1963).
16 April 1979
[10] E. Hylleraas, Z. Phys. 71(1931) 739. [11] W.C. Ermler and R.S. Mulliken, J. Chem. Phys. 66 (1977) 3031. [12] V.K. Nikulin and N.A. Gruschina, J. Phys. B11 (1978) 3553.
53_