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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 /

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

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.) xp

„=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

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

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

1.

2.

3.

4.

5.

Roothaan CCJ, Sachs L, Weiss AW (1960) Rev Mod Phys 32:186

Weiss AW (1961) Phys Rev 122:1826

Banyard KE (1968) J Chem Phys 48:2121

Baker CC, Banyard KE (1969) Phys Rev 188:57–62

AL-Khafaji KS (2005) A study of correction function to Hartree-Fock orbitals derived from

correlated wave function. PhD thesis, Al-Mustansiriyah university, Baghdad, Iraq

6. Coulson CA, Neilson AH (1961) Proc Phys Soc 78:831

7. Curl RF, Coulson CA (1965) Proc Phys Soc 85:647

8. Banyard KE, Seddon GJ (1973) J Chem Phys 58:1132

9. Seddon GJ, Banyard KE (1973) J Chem Phys 59:572

10. Boyd RJ (1975) J Chem Phys 53:592

11. Aman Alla SM (2007) Electron correlation for many atomic and ionic system. MSc thesis,

12. AL-Robayi EM (2002) A study of Coulomb hole for the ground state in momentum space for

He-like and Li-like ions. MSc thesis, College of Education (Ibn AL-Haitham), University of

13. Zielinski TJ, Barnett MP (2003) Development and dissemination of mathcad; see reference no.

44 therein

14. Lopez EP (1995) Physical chemistry: a practical approach. Williamstown, MA 01267

15. Avery J (1980) The quantum theory of atoms, molecules, and photons. Mc Graw Hill,

New York

10 Radial and Electron Correlation Effects for Helium and Some Helium Like Ions

169

16. Wiess AW (1963) J Chem Phys 39:1262

17. Clementi E, Roetti J (1974) Roothaan-Hartree-Fock atomic wave functions: basis functions

and their coefficients for ground and certain excited states of neutral and ionized atoms,

Z >D 54. At Data Nucl Data Tables 14:177

18. AL-Tamimei NC (2005) Calculation of effect of electronic correlation force on the energy of

some atoms. MSc thesis, College of Science for Women, Baghdad University, Baghdad, Iraq

19. McWeeny R, Sutcliffe BT (1969) Method of molecular quantum mechanics. Academic,

New York

20. Banyard KE, Mobbs RJ (1981) J Chem Phys 75(7):3433

21. Banyard KE, Al-Bayati KH (1986) J Phys B At Mol Phys 19:2211

22. AL-Meshhedany WA (2006) A study for nuclear magnetic shielding constant for Z D 2 to10.

MSc thesis, College of Science, Nahrain university, Baghdad, Iraq

23. Banyard KE, Baker CC (1969) J Chem Phys 51:2680

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,

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

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