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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 2075–2086
PII: S0953-4075(02)30783-1
Quantum fluid dynamics approach for electronic
structure calculation: application to the study of
ground-state properties of rare gas atoms
Amlan K Roy and Shih-I Chu
Department of Chemistry, University of Kansas, and Kansas Center for Advanced Scientific
Computing, Lawrence, KS 66045, USA
Received 14 November 2001, in final form 27 February 2002
Published 24 April 2002
Online at stacks.iop.org/JPhysB/35/2075
Abstract
We explore the usefulness of a quantum fluid dynamics (QFD) approach
for quantitative electronic structure calculations of many-electron systems.
By combining QFD and density functional theory, a single time-dependent
nonlinear QFD equation can be derived. The equation is further transformed
into a diffusion-type form by an imaginary-time evolution method, whose
asymptotic solution reaches a global minimum and the many-body groundstate wavefunction. The time-dependent generalized pseudospectral method
is extended to solve the diffusion equation in spherical coordinates, allowing
optimal and nonuniform spatial discretization and accurate and efficient solution
of the diffusion function in space and time. The procedure is applied to the
study of electronic energies, densities and other ground-state properties of
noble gas atoms (He, Ne, Ar, Kr, Xe). The results are in good agreement
with other best available values. The method offers a conceptually appealing
and computationally practical procedure for the treatment of many-electron
systems beyond the Hartree–Fock level.
1. Introduction
In recent years, density functional theory (DFT), based on the fundamental works of Hohenberg
and Kohn [1] and Kohn and Sham [2], has become a widely used formalism for electronic
structure calculations of atoms, molecules and solids [3]. The Kohn–Sham equations are
structurally similar to the Hartree–Fock (HF) equations, but include, in principle, exactly all
many-body effects through a local exchange–correlation (xc) potential. In this approach, the
electron density is decomposed into a set of N orbitals, leading to a set of N one-electron
Schrödinger-like equations to be solved self-consistently, where N is the total number of
electrons in the system. In this paper we consider an alternative approach based on the extension
of the hydrodynamic formulation of DFT.
0953-4075/02/092075+12$30.00
© 2002 IOP Publishing Ltd
Printed in the UK
2075
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A K Roy and S-I Chu
An approximate approach to many-particle systems was developed earlier by Bloch [4]
based on the framework of a time-dependent Thomas–Fermi (TDTF) model [5, 6]. The TF
model can be considered as a crude version of quantum fluid dynamics (QFD) where the
electronic system is considered as a gas of almost free electrons and the static electron densities
of many-electron systems can be calculated within a single equation. However, the dynamical
TF equations cannot be written as a single equation. More rigorous QFD formulations of
DFT were developed in the 1980s [7–9]. Of particular current interest is the development of a
single equation for the time-dependent electron density by the combination of QFD and DFT
approaches [10]. It has been shown that the two basic QFD equations, namely the equation
of continuity and the Euler-type equation of motion in 3D space (in terms of the two basic
variables, the electron density ρ(r , t) and the current density j (r , t)), can be combined to
obtain a single time-dependent generalized nonlinear Schrödinger equation (GNLSE). This is
different from the conventional DFT [3], self-interaction-free TDDFT [11–13] and TD current
DFT [14, 15] approaches where an individual occupied spin–orbital is treated explicitly. Such
a single-equation approach, in principle, allows the delivery of electron densities in the same
spirit of the Schrödinger equation, and maintains an accuracy beyond the HF level. Moreover,
it can significantly reduce the conceptual and computational difficulties for large systems,
even though at the expense of losing the orbital picture. The QFD–DFT–GNLSE approach
has recently been applied to the study of both static ground-state calculations [16, 17] and
dynamical processes [18–20] in cylindrical coordinates, using finite difference discretization
of coordinate spaces.
In this paper, we perform a precision calculation of the nonlinear QFD–DFT equation in
spherical coordinates by means of the time-dependent generalized pseudospectral (TDGPS)
technique [12, 21] developed recently, with an aim to assess the usefulness of the QFD–
DFT approach for realistic and quantitative investigation of electron structure calculations.
In order to calculate the ground-state properties of many-electron systems, the GNLSE is
first transformed into a diffusion-type equation by an imaginary time technique. The TDGPS
method allows nonuniform spatial grid discretization and has been shown to be capable of
providing high-precision time-dependent wavefunctions with the use of only a modest number
of grid points. The procedure has been applied successfully to the solution of time-dependent
Schrödinger and self-interaction-free TDDFT equations for the study of HHG processes of
the H atom [21, 22], rare gas atoms [11] and molecules [12], as well as for the calculation
of Rydberg-atom high-resolution spectroscopy [23, 24] in external fields. In this paper, we
extend the TDGPS method to the solution of the diffusion equation and the calculation of the
ground-state electronic properties of rare gas atoms.
The paper is organized as follows. In section 2, we outline the GNLSE formalism and the
relevant equations. The numerical procedure is presented in section 3. Detailed results and
discussions are given in section 4. This is followed by a conclusion in section 5.
2. Methodology
The QFD formulation of quantum mechanics was originally proposed years ago by
Madelung [25], de Broglie [26] and Bohm [27]. The QFD formulation requires solving
a set of nonlinear partial differential equations (PDEs) and thus is more complicated than
the linear Schrödinger equation. However, the QFD has a conceptually appealing feature;
namely, the electron cloud is treated as a classical fluid moving under the influence of classical
Coulomb forces and an additional quantum potential. Similar to the ab initio time-dependent
Schrödinger equation approach, the ab initio QFD approach for many-particular systems is
computationally formidable. The former involves the solution of 3N -dimensional PDEs with
QFD approach for electronic structure calculation
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time-dependent complex variables (wavefunctions), while the latter involves the solution of
6N-dimensional PDEs with time-dependent real variables (amplitudes and phases of complex
wavefunctions), where N is the number of electrons in the system. The combination of QFD
with DFT allows the reduction of this formidable problem to the solution of only one single
3D GNLSE which is computationally tractable. In this section, we outline the essence of this
GNLSE approach as well as the corresponding diffusion equation in imaginary time. More
detailed discussions can be found in [20].
Regarding all the electrons in an interacting many-electron system to be distributed over
the 3D space like a continuous classical fluid, the two basic QFD equations in terms of the
local variables, electron density, ρ(r , t) and current density, j (r , t), can be written as (atomic
units employed unless otherwise mentioned) [7, 10, 16–20]
(i) Continuity equation:
∂ρ(r , t)
(1)
+ ∇ · j (r , t) = 0,
∂t
(ii) Euler-type equation of motion:
δG[ρ] δEel –el [ρ]
∂χ (r , t) 1
(2)
+ (∇χ )2 +
+
+ v(r , t) = 0,
∂t
2
δρ
δρ
where j (r , t) = ρ∇χ (r , t), χ (r , t) being the velocity potential. Eel−el is the interelectronic
Coulomb repulsion energy (the Hartree term); G[ρ] is a universal density functional consisting
of kinetic and xc energy functionals and v(r , t) accounts for the TD potential including
electron–nuclear attraction and the interaction potential with the external field:
1
|∇ρ|2
dr + Tcorr [ρ] + Exc [ρ]. (3)
G[ρ] = Tw [ρ] + Tcorr [ρ] + Exc [ρ] =
8
ρ
Here Exc [ρ] is the xc energy functional, Tw denotes the Weizsäcker kinetic energy [3], which is
exact for one-electron and two-electron HF systems. Other than these two cases, the corrected
‘nonclassical’ kinetic energy term, Tcorr , is non-zero and, the exact form being unknown,
requires approximation:
Texact = Tw [ρ] + Tcorr [ρ].
Tcorr is the kinetic energy density functional providing the difference between the exact and the
Weizsäcker kinetic energy (see [16–20] and references therein). Within the rubric of DFT, this
form of the kinetic energy functional, keeping the Weizsäcker term as such plus a correction
term (often a TF-like term), is desirable, for it ensures proper local as well as global behaviour
and retains the shell structure (see [33], [3] pp 139–140 and references therein).
The entire time-evolving interacting system is described by the complex-valued
hydrodynamical wavefunction
(r , t) = ρ(r , t)1/2 eiχ (r,t) .
(4)
Now elimination of χ (r , t) from equations (1) and (2) results in the following TD QFD–DFT
equation of motion, the so-called GNLSE, viz:
1 2
∂(r , t)
.
(5)
− ∇ + veff ([ρ]; r , t) (r , t) = i
2
∂t
However, one can write equations (1) and (2) in imaginary time τ and substitute τ = −it, t
being the real time, to obtain
1 ∂ρ
+ ∇ · (ρ∇χ ) = 0,
(6)
−
i ∂t
1 ∂χ
δG[ρ] δEel−el [ρ]
1
(7)
= (∇χ )2 +
+
+ v(r , t).
i ∂t
2
δρ
δρ
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A K Roy and S-I Chu
After some simple algebra, followed by the elimination of χ (r , t), yields an equation which
closely resembles a diffusion-type equation:
1
∂R(r , t)
.
(8)
− ∇ 2 + veff ([ρ]; r , t) R(r , t) = −
2
∂t
The diffusion process is governed by the effective potential veff ([ρ]; r , t). It may be noted
that the diffusion function R(r , t) no longer resembles the hydrodynamical function (r , t) as
R(r , t) is not normalized at any time t (nonunitary) and does not directly correspond to the real
atomic/molecular system characterized by the veff ([ρ]; r , t). However, forcing normalization
of the diffusion function and evolution up to a sufficiently long time eventually leads to
the minimum-energy ground state in a global optimization scheme [16, 17]. veff ([ρ]; r , t)
comprises potentials of both classical and quantum origin:
δEel –el δEnu–el δExc δTcorr δEext
veff ([ρ]; r , t) =
+
+
+
+
.
(9)
δρ
δρ
δρ
δρ
δρ
The first three terms signify the inter-electronic repulsion, nuclear–electron attraction and xc
potentials, respectively, while the fourth term is the nonclassical correction term added to the
Weizsäcker kinetic energy. The last term arises from any interaction with the external field
(presently zero). Eel−el [ρ] and Enu−el [ρ] have the usual classical Coulombic forms. The exact
form of exchange energy functional has been used for He, while for other systems a simple
local energy functional form [28] has been employed. Thus, for He,
ρ(r , t)ρ(r , t)
1
dr dr ,
Ex = −
(10)
4
|r − r |
δEx
1
ρ(r , t)
=−
dr ,
(11)
δρ
2
|r − r |
while for other atoms,
ρ 1/3
LDA
ρ dr ,
−β
(12)
Ex = Ex
1 + r 2 ρ 2/3 /αx
where
ExLDA = −Cx
and
ρ 4/3 dr ,
(13)
4 1/3 2 r 2 ρ ρ + 3 αx
δExLDA
δEx
3
,
=
−β 2 2/3 2
δρ
δρ
1 + r αρx
(14)
δExLDA
4
= − Cx ρ 1/3 ,
δρ
3
(15)
with Cx = 43 (3/π )1/3 . β and αx are the two parameters to be determined empirically. On the
ground that at r → 0 or r → ∞ the energy densities of the two terms on the right-hand side of
equation (12) are identical, it is reasonable to assume β to be replaceable by Cx . This leaves
only one adjustable parameter, αx , for which a good choice has been 0.024 40 [28]. This form
of exchange functional shows correct asymptotic behaviour; it is local and, being gradient-free,
requires less computational effort. Earlier [17] this functional has been found to be capable of
yielding good-quality results for spherically symmetric systems. For example, the exchange
energies (in au) calculated from equation (12) are 1.026, 12.14, 30.15, 93.94 and 179.2 for He,
Ne, Ar, Kr and Xe, respectively; while the corresponding HF values are 1.026, 12.11, 30.19,
93.89 and 179.2, using the HF densities [29].
QFD approach for electronic structure calculation
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The simple local parametrized Wigner-type correlation energy functional [30] has been
used for all the systems. This functional has been quite successfully used for both ground and
excited (including autoionizing) states of atomic (see, e.g., [31, 32]), molecular systems, and
also for the laser–atom interaction processes in strong fields. Other justifications may be found
in [19]:
ρ
Ec = −
dr ,
(16)
a + bρ −1/3
δEc
a + cρ −1/3
=−
,
(17)
δρ
(a + bρ −1/3 )2
where a = 9.81, b = 21.437 and c = 28.582 667, respectively. Since the exact form of
Tcorr [ρ] is unknown, it must be approximated. Such an approximation [33] is provided by a
modified TF-like form, viz:
Tcorr = Ck f (r )ρ 5/3 (r ) dr ;
Ck = (3/10)(3π 2 )2/3 .
(18)
Here f (r ) is an r-dependent term satisfying the boundary condition f (r ) → 0 as r → 0 and
f (r ) → 1 as r → ∞, such that
total kinetic energy = Tw [ρ] + Tcorr [ρ],
δTcorr [ρ]
5
= Ck ρ 2/3 (r )g(r ),
δρ
3
where
g(r ) =
3
2
f (r ) + ρ −2/3 (r )
5
5
ρ(r )ρ̃(r )−1/3
(19)
(20)
δ ρ̃(r ) dr
δρ(r )
(21)
and ρ̃(r ) = ρ(r )f 3/2 (r ). For noble gas atoms, f (r ) and g(r ) may be expressed as a sum of
several Gaussian functions by making a semiempirical analogy [33]:
f (r) =
n
Ai exp[−αi (r − Ri )2 ]
i=1
g(r) =
n
(22)
Ai exp[−βi (r − Ri )2 ].
i=1
The values of Ai , Ri , αi and βi for Ne, Ar, Kr and Xe are given in [17].
3. Numerical solution of the diffusion equation: TDGPS method
The diffusion equation (8) can be rewritten as
L̂ R(r , t) = −
∂R(r , t)
= [Ĥ0 (r ) + V̂ (r , t)]R(r , t),
∂t
(23)
where L̂ denotes the nonlinear operator in square brackets and Ĥ0 is the ‘unperturbed’ operator
including the effective potential veff ([ρ]; r , t) in equation (9) at t = 0. V̂ is the interaction
potential with external fields (zero presently) and the remaining time-dependent effective
potential:
Ĥ0 (r ) = − 21 ∇ 2 + veff ([ρ]; r , 0),
(24)
V̂ (r , t) = veff ([ρ]; r , t) − veff ([ρ]; r , 0).
(25)
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A K Roy and S-I Chu
Now we extend the second-order split-operator scheme [21] in spherical coordinates in
energy representation for the time propagation:
R(r , t + +t) e−Ĥ0 +t/2 e−V̂ (r,θ,t++t/2)+t e−Ĥ0 +t/2 R(r , t) + O(+t 3 ).
(26)
Equation (26) shows that the time propagation of the diffusion function R(r , t) from t to t + +t
is achieved by three steps:
(i) First the wavefunction R(r , t) is propagated for a half-time step +t/2 in the energy space
spanned by Ĥ0 to obtain R1 (r , t) = e−Ĥ0 +t/2 R(r , t).
(ii) Then R1 (r , t) is transformed back to the coordinate space and propagated for a time step
+t under the influence of V̂ (r , t + +t/2) to obtain R2 (r , t).
(iii) Finally R2 (r , t) is transformed back to the energy space by Ĥ0 and propagated another
half-time step +t/2 to obtain R(r , t + +t).
Note that this time propagation scheme is different from other split-operator schemes
available in the literature [34, 35], where one usually chooses Ĥ0 to be the kinetic energy
operator. Advantages of using the energy representation have been explained elsewhere [21].
To achieve the time propagation, the diffusion function R(r , t) in spherical coordinates is
expanded in the Legendre polynomial basis, Pl (cos θj ):
gl (ri , t) Pl (cos θj ),
(27)
R(ri , θj , t) =
l
where Pl ’s are the normalized Legendre polynomials. gl (ri , t) can be determined by the
Gauss–Legendre quadratures:
gl (ri , t) =
L+1
wk Pl (cos θk )R(ri , θk , t),
(28)
k=1
where {cos θk } denote the L + 1 zeros of PL+1 (cos θk ) and {wk } are the corresponding
quadrature weights. The propagation in the energy space (step (i) in equation (26)) can now
be accomplished through
0
[e−Ĥl +t/2 gl (ri , t)]Pl (cos θj ),
(29)
e−Ĥ0 +t/2 R(ri , θj , t) =
l
with
1 d2
l(l + 1)
+
+ veff ([ρ]; r , 0).
(30)
2
2 dr
2r 2
Note that in equation (29) each partial-wave diffusion function component gl is propagated
independently under individual Ĥl0 energy space, leading to efficient propagation of the total
diffusion function in step (i). The key step is to map the infinite domain [0, ∞] or [0, rmax ] to
[−1, 1] through a nonlinear mapping r = r(x) [36]:
1+x
.
(31)
r = r(x) = L
1−x+α
This allows for denser grids near the origin and a considerably smaller number of grid points
suffice to achieve accurate results in contrast to the equal-spacing methods where a considerably
larger number of grid points are required. Here, L and α = 2L/rmax are the mapping
parameters. Finally, following a symmetrization procedure [36]:
(32)
φl (r) = r (x) χl (r(x)),
Ĥl0 = −
one can rewrite the operator in the symmetrized form as
Ĥl0 (r) = −
1 1 d2 1
+ Vl (r(x)),
2 r (x) dx 2 r (x)
(33)
QFD approach for electronic structure calculation
2081
where
l(l + 1)
+ veff ([ρ]; r , 0).
(34)
2r 2
A key step in the time propagation of equation (29) is to construct the evolution operator
−Ĥl0 +t/2
e
≡ S(l) through an accurate and efficient representation of Ĥl0 . Here we extend the
generalized pseudospectral (GPS) method [36] to achieve optimal grid discretization and an
accurate solution of the eigenproblem of Ĥl0 . For example, in the earlier study of the Dirac
equation [36], it was found that 20 radial grid points are sufficient to achieve 10–14 digits of
accuracy for the first few eigenvalues of He+ .
In the hydrodynamical approach using imaginary time propagation, the initial state can
be, in principle, any arbitrary function. As the diffusion equation propagates in time, it will
converge to the many-body ground state. In the present work, we are particularly interested
in the exploration of the improvement of the QFD–DFT single GNLSE approach in electron
structure calculation beyond the HF level. Thus we choose the HF wavefunction as our initial
state at t = 0. Note that, in the diffusion equation time propagation, the function R(t) is
not normalized as time propagates. Thus R(t) needs to be renormalized after each time step
propagation. Then the difference of the expectation values of the nonlinear operator L̂ at two
successive time steps, say at t = n+t and (n + 1)+t, is calculated:
Vl =
+2 = L̂n+1 − L̂n ,
(35)
L̂n+1 = R n+1 (r , t)|L̂|R n+1 (r , t).
(36)
where
As time propagates, +2 is getting smaller until it becomes less than a predefined tolerance
limit (10−10 in the present case). In the present study, 200–400 grid points are used for the
GPS discretization of the radial coordinates r and +t = 0.01–0.1 au is used in the time
propagation to achieve convergence. This should be compared with other previous imaginary
time propagation methods [17] using finite difference discretization in spherical coordinates,
the latter requiring a considerably larger number of grid points (e.g. 5001 points were used
in [17]) and a smaller time step (typically +t = 0.002–0.0005). Thus the TDGPS method
provides a powerful numerical technique for the solution of diffusion equations, which is
computationally orders of magnitude more efficient than the equal-spacing time-dependent
techniques. In the following section, we show that the results of electronic structure calculations
using the TDGPS procedure are also considerably better.
4. Results and discussion
In this section we present the nonrelativistic electronic structure calculations of the ground
states of rare gas atoms using the procedure described in the last two sections. Table 1
summarizes the main results for He, Ne, Ar, Kr and Xe. The first row of each entry shows
the present results. The results from the previous finite-difference calculations [17] are shown
below the present results in parentheses whenever they are available. The corresponding HF
values and the best available results (denoted by ‘others’ in the table) from the literature are
listed for comparison. For all the rare gas atoms considered, we found the present results of the
total electronic energies are considerably better than the HF values and are in close agreement
with those best available results. The present results are also significantly improved over
the previous finite difference calculations [17], particularly for He, Ne and Ar. The Kr and
Xe results can be further improved if the grid structure is further optimized and more grid
points are used. We note that the ‘exact’ total energy results for Kr and Xe are not available
and the values listed here are obtained by adding the second-order many-body perturbation
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A K Roy and S-I Chu
Table 1. Calculated ground-state properties of He, Ne, Ar, Kr and Xe (in au) along with
literature data for comparison. Numbers in parentheses denote results from the finite-difference
calculations [17].
He
Ne
Ar
Kr
Xe
HFa
Others
2.9031
(2.8973)
2.8617
2.9037b,j
128.9103
(128.9065)
128.5470
128.938c,j
527.5710
(527.5486)
526.8174
527.604d,j
2753.8832
(2753.8809)
2752.0546
2753.8896e
7234.9815
(7234.9742)
7232.1302
7235.0512e
Exactf
1.0273
(1.0325)
1.026
12.1272
(12.1111)
12.11
29.5584
(29.4850)
30.19
91.7238
(91.5847)
93.89
174.1789
(173.9435)
179.2
Others
0.0434
(0.0423)
0.042f,j
0.3578
(0.3561)
0.390f,j
0.7102
(0.7011)
0.787f,j
1.7645
(1.7529)
1.835g
2.8558
(2.8407)
2.921g
2.9030
(2.8973)
HFh
94.2325
(94.2068)
90.6140
322.1798
(322.0345)
308.4206
1377.7633
(1377.5940)
1276.7349
3227.1387
(3226.9174)
2932.0548
HFh
34.6769
(34.7006)
37.3886
205.3967
(205.5177)
214.4033
1375.9691
(1376.3217)
1465.2484
4007.9044
(4008.3670)
4298.9068
−E
−Ex
−Ec
Tw
Tcorr −V /T HFa
2.0000
1.9999
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
1.9999
2.0000
HFi
6.0471
5.9955
42.7056
41.4890
84.1014
81.3908
186.0326
175.8599
294.1121
274.4421
HFi
1.6993
1.6873
3.1115
3.1113
3.8475
3.8736
5.0473
5.0792
5.8496
5.8866
HFi
0.9188
0.9273
0.7957
0.7891
0.9717
0.8928
0.7666
0.7289
0.7495
0.7233
HFi
1.1613
1.1848
0.9733
0.9372
1.7401
1.4464
1.1809
1.1722
1.2483
1.1602
r −2 r −1 r
r 2 a
[29].
[44].
c [45].
d [46].
e Adding MBPT CI correlation energy [37] to the HF energy [29].
f [38].
g [37].
h As quoted in [17].
i [42].
j Exact results.
b
theory (MBPT) correlation energy results [37] to the HF energies [29]. It is gratifying that the
TDGPS approach can be used to perform high precision calculation of the diffusion equation
with the use of only a modest number of grid points.
An analysis of the results for individual exchange and correlation energies in table 1 is given
below. The exchange energies (Ex ) of the present calculations show a satisfactory agreement
with the HF results [38]. For He and Ne, the calculated exchange energy is nearly exact,
while for Ar, Kr and Xe, there is an underestimation by 2.09–2.80%. This indicates that the
simple local exchange functional Ex [28] in equation (12) is reasonably accurate, though not as
accurate as the more elaborate nonlocal gradient-corrected functionals such as those of Perdew
and Yue [39] and Becke [38], which show a closer agreement with HF exchange energies.
QFD approach for electronic structure calculation
2083
60
50
Kr ground state
Radial density
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
r(a.u.)
Figure 1. Radial density plot of Kr (in au).
The ‘exact’ correlation energies are available only for He, Ne and Ar [38]; for Kr and Xe,
the best available results [37] are cited in table 1 for comparison. The simple Wigner-type
local correlation energy functional seems to be reasonably good for the systems considered.
For He, it is nearly exact, otherwise underestimated by about 2.23–8.25%; Ar being the worst
case. Compared with other generalized-gradient approximations (GGA), Perdew’s GGA [40]
correlation energy functional gives better results for Ne and Ar but worse results for He, Kr and
Xe. On the other hand, the Lee–Yang–Parr’s (LYP) GGA correlation energy functional [41]
gives better results for all the cases. We note that the primary purpose of this work is to explore
the feasibility of extending the TDGPS to the solution of the QFD–DFT single nonlinear
equation with imaginary time propagation and not for the detailed comparison of various
energy functionals. GGA-type xc energy functionals can be easily adopted in the present QFD
approach. It is nevertheless instructive to assess the reliability of these local exchange and
correlation energy functionals, since they are relatively simple in form and can be valuable for
the extension to larger systems and time-dependent processes [43].
Table 1 shows that the virial theorem is well satisfied for all the cases. The Weizsäcker
term (Tw ) and the correction term Tcorr contribute about 73.10–44.60% and 26.90–55.40% to
the total kinetic energy, respectively. With an increase in the nuclear charge Z, Tw becomes
smaller and Tcorr larger; for Kr the two terms contribute nearly equally. Tw is always positive,
and Tcorr is essential for the single equation to deliver accurate results. While the exact form
of Tcorr is not available, its approximate form may be constructed from the electron density
obtained from other more sophisticated ab initio CI or Monte Carlo calculations. The good
results obtained from the approximate Tcorr forms [33] used in the present paper for the rare gas
atoms show that this procedure appears to be feasible and promising. However, the search for a
more general Tcorr form is very desirable and valuable if the single time-dependent QFD–DFT
equation approach is to be extended to other larger systems in the future.
Figures 1 and 2 show the radial density plots for Kr and Xe. We note that the radial densities
calculated maintain the expected shell structure and closely resemble the HF densities (not
shown). This may be accounted for by the choice of the kinetic energy functional, where f (r)
2084
A K Roy and S-I Chu
90
80
Xe ground state
70
Radial density
60
50
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
r(a.u.)
Figure 2. Radial density plot of Xe (in au).
800
Ar
Kr
700
600
δ Tcorr/δ ρ
500
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
r(a.u.)
Figure 3. Plot of δTcorr [ρ]/δρ against r, for Ar and Kr, in au
and g(r) functions contain peaks and valleys and closely reflect the atomic shell structures—
the maxima of f (r) corresponding to the minima in the radial density. Figure 3 depicts the
potential δTcorr [ρ]/δρ for Ar and Kr. This clearly shows that the nonclassical correction term
in the kinetic energy functional, Tcorr [ρ], is essential to include for the accurate calculation of
many-electron systems through a single QFD equation. The expectation values of the single-
QFD approach for electronic structure calculation
2085
particle operators (density normalized to unity), r n , n = −2, −1, 1, 2, which determine the
size of the atoms and are related to other atomic properties such as nuclear magnetic shielding
(1/r) and diamagnetic susceptibility (1/r 2 ), are also tabulated in the table. These results
are also quite close to the best available literature values. The HF values [41] are cited here
for comparison.
5. Conclusion
In this paper, we show that nonrelativistic electronic energies, densities and other ground-state
properties of rare gas atoms can be calculated by means of a single time-dependent QFD–DFT
equation and an imaginary time evolution technique. The TDGPS method allows an efficient
and accurate solution of the resulting diffusion equation in space and time. The calculated
electronic energies and other properties are considerably better than the HF values and are
in good agreement with the best available results in the literature. The approach holds the
promise of offering a practical route to larger systems, bypassing the ab initio many-electron
wavefunction. However, the success will rely mostly on the availability of accurate kinetic
energy and xc energy functionals. Extension of the approach to the study of multiphoton and
highly nonlinear optical processes, such as high-order harmonic generation in intense laser
fields, is currently in progress [43].
Acknowledgments
This work is partially supported by National Science Foundation under contract no PHY0098106. We acknowledge Kansas Center for Advanced Scientific Computing for the use of
Origin2400 supercomputer facilities sponsored by NSF-MRI program no DMS-9977352.
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