15 The Inter-Particle Distribution Function f(r12) in HF Method
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162
K.H. AL-bayati and E.F. Saeed
10.18 Partial Distribution Function g.r12 ; r1 /
The partial distribution function g.r12 ; r1 / represents the probability of finding an
inter-electronic separation r12 given by the relationship
Z
g.r12 ; r1 / D 8
2
r1 r12
jr12 Cr1 j
jr12 r1 j
0 .r1 ; r2 /r2 d r2
(10.35)
This function depends upon both electronic separation (r12 ) and the position of
the first electron. The function g.r12 ; r1 / is useful to illustrate the behavior of the test
electron “reference electron”, and the change in distance occurs when the location
of electron 1 from nucleus is fixed. This function is used to study the partial Fermi
and Coulomb holes.
10.18.1 Partial Distribution Function g.r12 ; r1 / in HF Method
For the K-shell by substituting Eq. 10.14 in Eq. 10.35 the result is [11]
Z
HF
.r12 ; r1 / D 0:5r1 r12
gk.1
s/
jr12 Cr1 j
jr12 r1 j
2
2
R1s
.r1 /R1s
.r2 /r2 d r2
(10.36)
10.18.2 The Partial Distribution Function g.r12 ; r1 /
in CI Method
For the K-shell by substituting Eq. 10.16 in Eq. 10.35 the result is
Z
g.r12 ; r1 / D 0:5r1 r12
jr12 Cr1 j
jr12 r1 j
h
i
R'2 .1/R2 .2/ C R'2 .2/R2 .1/ r2 d r2
(10.37)
10.18.3 Calculation Method of Partial Coulomb Hole g.r12 ; r1 /
The partial Coulomb hole g.r12 ; r1 / allows us to analyze the effect of the coulomb
hole when the test (reference) electron say particle 1 at specific radial distance from
the atomic nucleus [11] g.r12 ; r1 / is defined as
g.r12 ; r1 / D gCI .r12 ; r1 /
gHF .r12 ; r1 /
(10.38)
10 Radial and Electron Correlation Effects for Helium and Some Helium Like Ions
163
The partial distribution function g.r12 ; r1 / for both CI and HF wave function
K-shell is given by
Z
gCI;HF .r12 ; r1 / D 0:5r1 r12
jr12 Cr1 j
jr12 r1 j
0
CI;HF
.r1 ; r2 /r2 d r2
(10.39)
10.19 Results and Discussion
10.19.1 The Two-Particle Radial Density Distribution D.r1 ; r2 /
The two-particle radial density distribution function D.r1 ; r2 / is particularly sensitive to radial correlation. Table 10.1 shows that as the atomic number Z increases,
the two-particle radial density distribution function D.r1 ; r2 / is increased at r1 D r2
and because of the symmetry state of K-shell, the maximum values of D.r1 ; r2 / are
always located along diagonal when r1 D r2 . The effect of electron correlation upon
the electron density shows that the location and the maximum value of two-particle
density distribution function DHF .r1 ; r2 / is greater than DCI .r1 ; r2 /.
10.19.2 The One Particle Radial Distribution Functions D.r1 /
Table 10.2 shows the maximum value of the one particle radial densities distribution
function D.r1 / and its location for the considered atomic system in HF and CI
approximations. From this table the following notes can be found: The one particle
density D.r1 / in both wave functions increases with increasing the atomic number
Z, whereas the values of locations r1 decrease with increasing the atomic number Z
Table 10.1 Maximum value of the two-particle radial density distribution function and the location for He-series in the Hartree-Fock (HF) wave function and
configuration-interaction (CI) wave function
Atom or ion Wave function r1 D r2 D rk
Max. D.r1 ; r2 / Radial corr. %*
He
HF
0.569
0.750
93.866
HF [18]
0.575
0.75
CI
0.521
0.704
LiC
HF
0.363
1.981
85.714
HF [18]
0.36
1.981
CI
0.343
1.698
BeCC
HF
0.266
3.796
83.219
HF [18]
0.27
3.796
CI
0.256
3.159
All results are expressed in a.u
*Radial corr. % D .DCI .r1 ; r2 /=DHF .r1 ; r2 // 100
164
K.H. AL-bayati and E.F. Saeed
Table 10.2 The one particle radial densities D.r1 / in HF and CI approximations
for K-shell of the He-Like atoms and the maximum locations r1
Atom or ion
Wave function
r1
Max D.r1 /
Radial corr. %
He
HF
0.5692
0.8663
97.12
HF [18]
0.5692
0.8663
CI
0.5217
0.841
LiC
HF
0.3634
1.407
93.24
HF [18]
0.3634
1.4074
CI
0.3434
1.312
BeCC
HF
0.2668
1.949
92.25
HF [18]
0.2667
1.9487
CI
0.2562
1.804
Table 10.3 The one particle expectation value hr1k i and standard deviation for both HF and CI
approximation
hr1k i
Atom or ions
He
LiC
BeCC
Wave function
HF
HF [22]
CI
CI [23]
HF
HF [22]
CI
CI [23]
HF
HF [22]
CI
hr1 2 i
5:995
5:981
6:438
6:018
14:910
14:912
15:456
14:929
27:829
27:829
28:335
hr1 1 i
1.687
1.68
1.702
1.688
2.687
2.687
2.668
2.687
3.680
3.687
3.641
hr10 i
1
1
1
1
1
1
1
1
1
1
1
hr11 i
0.927
0.919
0.945
0.929
0.572
0.572
0.596
0.572
0.414
0.414
0.433
hr12 i
1.184
1.169
1.254
1.192
0.445
0.444
0.502
0.446
0.231
0.231
0.264
hr13 i
1.94
2.180
0.439
0.559
0.163
0.212
r1
0.570
0.569
0.625
0.343
0.342
0.408
0.245
0.245
0.295
as a result from the attraction force of the nucleus to the electrons. Also for all Z,
the maximum value of D.r1 / in CI wave function is less than in HF wave function
due to radial correlation effect.
10.19.3 One-Particle Expectation Value hr1k i
Table 10.3 shows the following statements for all atom and ions under investigation.
For both HF and CI wave functions, the largest value of hr11 i is in the He-atom,
because it has the smallest atomic number with respect to other ions, leading to
smallest Coulombs attraction between the nucleus and the electron in the K-shell. As
the atomic number is increased the Coulombs attraction force is also increased. This
leads to decrease of the one particle expectation value which means that moving
region for electrons will be smaller and smaller with increasing atomic number
Z and the uncertainty (i.e.) xp
„=2 of finding the electronic position will
10 Radial and Electron Correlation Effects for Helium and Some Helium Like Ions
165
Table 10.4 The two-particle expectation values hr1k r2k i in both HF and CI wave functions
hr1k r2k i
Atom or ions
He
LiC
BeCC
Wave function
HF
CI
HF
CI
HF
CI
hr1 2 r2 2 i
35:946
39:049
222:3
247:7
774:4
845:2
hr1 1 r2 1 i
2:846
3:021
7:368
7:223
13:543
13:597
hr10 r20 i
1
1
1
1
1
1
hr11 r21 i
0.859
0.883
0.327
0.369
0.172
0.196
hr12 r22 i
1.403
1.615
0.198
0.267
0.054
0.073
hr13 r23 i
3.765
4.945
0.192
0.293
0.027
0.042
decrease and the certainty of finding r1 will be increased. It is obvious that r1 in
CI is greater than in HF approximations because the wave function of CI has more
terms than HF wave function.
10.19.4 The Two Particle Expectation Value
The amounts of the two-particle expectation values hr1k r2k i have been evaluated
and tabulated in Table 10.4. One may find the following notes: (1) For both wave
functions the largest value of hr11 r22 i is in the He-atom, and this result is similar to
that found in hr11 i for the same region. (2) Two particle expectation values for CI
approximation are greater than HF approximation because of the radial correlation
of the two particle radial distribution function. The two particle expectation value is
the square of the one particle expectation value for both HF and CI approximations
because of the symmetry.
10.19.5 Function f .r12 / for He-Like Atoms
The probabilities of the inter particle distribution function f .r12 / for both HF and
CI wave functions are tabulated in Table 10.5 for He-like atoms. The results show
that the maximum value of the inter particle distribution function for both HF and
CI approximation is increased and is closer to the nucleus as the atomic number
(Z) increases because the distance between electrons r12 decreases. This behavior
shows that the K-shell density shrinkage toward the nucleus due to the Coulomb
repulsion force will be greater than the Coulomb attraction force. In each Z, the
inter particle distribution function in fHF .r12 / is greater than in fCI .r12 /, and the
radial correlation percent for f .r12 / is 98.9%, 91.7% and 88.9% for Z D 2, 3 and 4
respectively. The value of r12 such that f .r12 / D 0 may be interpreted as the radius
of Coulomb hole in Table 10.6. The result of He-like atoms shows that the addition
of radial correlation has reduced, as atomic number Z increases, the depth increases
while the radius decreases.
166
K.H. AL-bayati and E.F. Saeed
Table 10.5 The inter particle distribution function f .r12 / in maximum location for
the inter particle distance
Atom or ions
Wave Function
r12
Max. f .r12 /
f .r12 /
z
Radial corr. %
He
HF
HF [22]
CI
HF
HF [22]
CI
HF
HF [22]
CI
0.995
0.995
0.921
0.623
0.623
0.580
0.455
0.454
0.423
0.629
0.629
0.622
1.029
1.029
0.943
1.439
1.427
1.279
0.314
0.314
0.311
0.343
0.343
0.314
0.359
0.356
0.319
98.9
LiC
BeCC
91.7
88.9
Table 10.6 The radius of Coulomb hole r12 and its area f .r12 /
Atom or ions Radius of coulomb hole Area of the coulomb hole
He
1.198
0.008
LiC
1.195
0.055
0.845
0.07
BeCC
Table 10.7 The inter-particle expectation values
for He-like atoms and the standard deviation
k
hr12
i
˝ 2 ˛ ˝ 1 ˛
r 12
Atom or ions
r 12
He
HF
1.842
1.025
HF [18] 1.8421
1.0258
CI
1.815
1.017
LiC
HF
4.726
1.651
HF [18] 4.7264
1.6517
CI
4.473
1.583
BeCC
HF
8.934
2.261
HF [18] 8.944
2.277
CI
8.311
2.158
˝ k ˛
r 12 of the HF and CI approximation
˝
r 0 12
1
1
1
1
1
1
1
1
1
˛
˝ 2 ˛
r 12
1.362
1.3621
1.396
0.838
0.8381
0.907
0.609
0.6058
0.663
˝ 1 ˛
r 12
2.3695
2.3697
2.549
0.8906
0.8906
1.0791
0.4681
0.4637
0.5729
r12
0.717
0.7171
0.774
0.433
0.4334
0.506
0.310
0.3109
0.364
All values are in a.u
10.19.6 Inter Particle Expectation Value r k and Standard
12
Deviation
k
The inter-particle expectation values hr12
i of the HF and CI approximation for Helike atoms have been tabulated in Table 10.7. One may conclude that the different
k
regions of f .r12 / will be emphasized by the function hr12
i when k takes positive, and
negative values ( 2 Ä k Ä 2). A particularly useful concept displaying the spread
of position density f .r12 / about inter particle distance is the standard deviation
k
r12 . For each Z the inter-particle expectation values hr12
i increase when electrons
10 Radial and Electron Correlation Effects for Helium and Some Helium Like Ions
167
Table 10.8 The maximum location of the partial distribution function g.r12 ; r1 /for
both the HF and CI wave functions and the maximum location for r12 when r1D rk
Atom or ions
Wave function
r12
r1D rk
Maximum g.r12 ; r1/
He
HF
0.7575
0.569
0.761
CI
0.6912
0.521
0.739
Li
HF
0.4843
0.363
1.991
CI
0.4385
0.343
1.798
BeCC
HF
0.3578
0.266
3.819
CI
0.3264
0.2562
3.338
All results are expressed in a.u
Table 10.9 The maximum location of the partial distribution function g.r12 ; r1 / for
both the HF and CI wave functions and the maximum location for r1 when r12 D rk
Atom or ions
Wave function
r1D rk
r1
Maximum g.r12 ; r1/
He
HF
0.995
0.685
0.690
CI
0.921
0.693
0.681
Li
HF
0.623
0.363
1.991
CI
0.580
0.438
1.648
HF
0.455
0.312
3.523
BeCC
CI
0.423
0.316
3.089
are located in the region near the nucleus (when k goes from 1 to 2), while
k
hr12
i increases (when k goes from 1 to 2) for (He, LiC ) and decreases for (BCC )
because the electron cloud is further away than the nucleus. As Z increases the
k
k
k
inter-particle expectation values hr12
i decrease, and hr12
iCI is greater than hr12
iHF
at negative values of k whereas
at
positive
values
of
k
we
find
opposite
behavior
˛
˝
k
that means the values of r k 12 HF are greater than hr12
iCI . The standard deviation
results indicate that each f .r12 / distribution function has become less diffuse when
electrons correlated and it has therefore sharpened up about its maximum. As Z is
increased r12 decreases for both HF and CI wave functions.
10.19.7 Partial Distribution Function g.r12 ; r1 /
The influence of the correlation in partial distribution function g.r12 ; r1 / of r12 and
r1 for Z D 2, 3 and 4 in the K-shell is shown in Tables 10.8 and 10.9 for the HF and
CI wave functions. We notice that as Z increases the maximum value g.r12 ; r1 / for
both HF and CI approximations increases because the Coulomb attraction forces is
greater than the Coulomb repulsion forces in positive ions so the probability density
of finding pair electrons in the region near the nucleus in positive ions is larger than
that in He-atom. As expected gCI .r12 ; r1 / is less than gHF .r12 ; r1 / because in HF the
approximation each electron move independently of each other, so this hypothesis
neglects the details of the electronic repulsion which will reduces the HF results,
while CI approximation takes it in account. The g.r12 ; r1 / surface shows that the
maximum density is always located along the diagonal such that r12 > r1 :
168
K.H. AL-bayati and E.F. Saeed
10.20 Conclusions
From the present work, all atomic properties obtained with HF and CI wave
functions are Z dependent. As the atomic number Z increases, for both approximations the one, two-particle radial density distribution function D.r1 /, D.r1 ; r2 /,
the electron density at rk , the inter particle distribution function f .r12 / and the
partial distribution function g.r12 ; r1 / are increased. For both HF and CI wave
1
functions, the largest value of hr11 i, hr11 r21 i and hr12
i is in the He-atom, because
it has the smallest atomic number with respect to other ions. This leads to smallest
Coulombs attraction between the nucleus and the electron in the K-shell for this
ground state element (Z goes from 2 to 4). The inter particle distribution function
and the magnitude density of g.r12 ; r1 / for both approximations are increased
and are closer to the nucleus as the atomic number (Z) is increased and the
distance between electrons r12 is decreased. This behavior shows the K-shell density
shrinkage toward the nucleus because the Coulomb repulsion force will be less
than the Coulomb attraction force. The one, two-particle, inter-particle and partial
distribution functions are systematically shifted for He-like atoms when they are
plotted as functions of the atomic number. The radial correlation of one, twoparticle radial density distribution functions D.r1 ; r2 /and D.r1 / are increased
as Z increases and closer to the nucleus. The Coulomb hole and partial Coulomb
hole are increased as Z increases, whereas the radius of Coulomb hole is decreased
and the depth is increased Z increases.
References
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College of Education (Ibn AL-Haitham). University of Baghdad, Baghdad, Iraq
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Baghdad, Bagdad, Iraq
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16. Wiess AW (1963) J Chem Phys 39:1262
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some atoms. MSc thesis, College of Science for Women, Baghdad University, Baghdad, Iraq
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Chapter 11
Dynamical Role of the Fictitious Orbital Mass
in Car-Parrinello Molecular Dynamics
Sheau-Wei Ong, Eng-Soon Tok, and H. Chuan Kang
Abstract We investigate ion-orbital interaction in Car-Parinnello molecular
dynamics (CPMD) analytically and numerically in order to probe the role of the
fictitious orbital mass. We show analytically that this interaction can be described
by linearly coupled oscillators when the system is sufficiently close to the ground
state. This leads to ionic vibrational modes with frequency ăM that depends upon the
ionic mass M and the orbital mass as !M D!0M Œ1 C. =M / in the limit of zero
/M; ă0M is the Born-Oppenheimer ionic frequency and C depends upon the ionorbital coupling force constants. This analysis provides new insight on the orbital
mass dependence of the dynamics, and suggests a rigorous method of obtaining
accurate ionic vibrational frequency using CPMD. We verify our analytical results
with numerical simulations for N2 , and discuss in detail the dynamical interaction
between the ionic and the fictitious orbital modes in CPMD. Our results demonstrate
that displacement from the ground state significantly affects ionic frequencies. In
the linear regime this results in the linear dependence of ionic vibrational frequency
upon /M. In the non-linear regime, even the ionic geometry deviates from the
correct ground-state structure, highlighting the importance of staying close to the
ground state in CPMD calculations.
S.-W. Ong • H.C. Kang ( )
Department of Chemistry, National University of Singapore, 3 Science Drive 3,
117543 Singapore, Singapore
e-mail: chmosw@nus.edu.sg; chmkhc@nus.edu.sg
E.-S. Tok
Department of Physics, National University of Singapore, 3 Science Drive 3, 117543 Singapore,
Singapore
e-mail: phytokes@nus.edu.sg
M.G. Bhowon et al. (eds.), Chemistry for Sustainable Development,
DOI 10.1007/978-90-481-8650-1 11, © Springer ScienceCBusiness Media B.V. 2012
171
172
S.-W. Ong et al.
11.1 Introduction
The Car-Parrinello molecular dynamics (CPMD) method [1] initiated the field
of ab initio molecular dynamics and is an important technique applied to a
wide range of problems. Originally introduced to investigate problems involving
electronic structure in condensed matter within the framework of density functional
theory it has been applied to other ways of solving electronic structure problems
[2, 3], and also to other problems [4, 5] where the forces acting on the interesting
degrees of freedom are determined by underlying fast degrees of freedom. Within
density functional theory applications the central idea in CPMD is the simultaneous
propagation of the Kohn-Sham (KS) orbital degrees of freedom along with the ionic
degrees of freedom. This is achieved by treating the orbitals as a set of generalized
classical displacements each with an associated fictitious mass , thereby reducing
the problem to one where Newtonian dynamics is used to relax the ionic positions
and the electronic structure simultaneously [1]. This introduces fictitious ion-orbital
interactions that affect the real dynamics of the ions because during a CPMD
simulation the orbitals are not exactly on the Born-Oppenheimer surface, but
oscillate about it. It has been shown that the error in the CPMD trajectory relative
to the true trajectory on the Born-Oppenheimer surface is bounded by , and thus
the adiabatic limit is reached when goes to zero. In calculations the choice of
, necessarily finite, is made by balancing the computational cost and the accuracy
both of which increase as decreases.
The basic idea is that although the forces at any instant in a CPMD calculation are
different from the forces calculated using the ground state electronic structure, these
instantaneous differences average to zero over a time-scale relevant to investigations
of ionic dynamics because the fictitious orbital dynamics are much faster [6] and
their fluctuations have much shorter time-scales [7, 8]. Although the CPMD method
has been successfully and widely used, and the important work in Ref. [6] has
provided a heuristic understanding of the dynamics, the current understanding of the
interaction between the ionic and the orbital degrees of freedom is not satisfactory.
In particular, recent work [9, 10] demonstrates that the forces in CPMD, even when
averaged over fluctuations due to the orbital motion, are not equal to the groundstate forces. It is commonly held that the motion of the ions in the field due to the
rapidly fluctuation electronic structure effectively rescales the mass of the ions. This
mass rescaling can be estimated within a rigid-ion approximation which is rigorous
only for weakly-interacting ions.
Even with rescaled ionic masses taken into consideration in calculating the ion
trajectories, it was found that the forces on the ions are still not correctly calculated
[10]. This is problematic since it implies that the ionic forces in CPMD deviate from
the actual forces in ways that are not understood currently, even though it is pointed
out that the errors are systematically reduced when the orbital mass is decreased [9].
In addition, significant systematic drift in the fictitious electronic kinetic energy is
observed in simulations when is large. For instance, in liquid water calculations