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The self-consistent Kohn-Sham equations for many-electron atoms are solved using the Coulomb wave function Discrete Variable Method (CWDVR). Wigner type functional is used to incorporate correlation functional. The discrete variable method is used for the uniform and optimal spatial grid discretization and solution of the Kohn-Sham equation. The equation is numerically solved using the CWDVR method. First time we have reported the solution of the Kohn-Sham equation on the ground state problem for the many-electronic atoms by the CWDVR method. Our results suggest CWDVR approach shown to be an efficient and precise solution of ground-state energies of atoms. We illustrate that the calculated electronic energies for He, Li, Be, B, C, N and O atoms are in good agreement with other best available values.

Numerical approach of many-electron systems is extremely difficult computation. Density functional theory (DFT) [

To the best of our knowledge, no one reported the solution of the Kohn-Sham equation on the ground state problem for the many-electronic atoms by the CWDVR method. Furthermore, we accurately present the ground state energies of the many-electronic atoms by the CWDVR method.

This paper consists of methodology and results obtained within the CWDVR method. We show that ground state energy values calculated by the present method are in good agreement with other precise theoretical calculations. Finally, we present conclusions. Here, atomic units (a.u.) are used throughout this paper unless otherwise specified.

In this section, we first give a brief introduction to the DVR constructed from orthogonal polynomials and Coulomb wave functions, which will be used to solve the Kohn-Sham equation for many-electron atomic systems. The DVR approach basis functions can be constructed from any complete set of orthogonal polynomials, defined in the domain with the corresponding weight function [

It is known that a Gaussian quadrature can also be constructed using nonclassical polynomials. The DVR derived from the Legendre polynomials has been shown by Machtoub and Zhang [

It is known that a Gaussian quadrature can also be constructed using nonclassical polynomials. The DVR derived from the Legendre polynomials has been shown by Machtoub and Zhang [

The time dependent single particle Kohn-Sham equation has the form

i ∂ ψ j ( r , t ) ∂ t = ( H ⌢ 0 + υ e f f ) ψ j ( r , t ) , j = 1 , N ¯ (1)

here, ψ ( r , t ) is the single particle Kohn-Sham orbit of N electron atom, H ⌢ 0 —atomic Hamiltonian, υ e f f is the time dependent effective potential, and charge density depends on the coordinates and time and is given by

ρ ( r ) = ∑ j = 1 N | ψ j ( r ) | 2 (2)

However, one can write Equation (1) in imaginary time τ and substitute τ = − i t , t being the real time, to obtain a diffusion-type equations:

− ∂ R j ( r , t ) ∂ t = ( − 1 2 ∇ 2 + υ e f f ) R j ( r , t ) . (3)

The Kohn-Sham effective local potential contains both classical and quantum potentials and can be written as:

υ e f f [ ρ ; r , t ] = δ E e e δ ρ + δ E n e δ ρ + δ E x c δ ρ + δ E e x t δ ρ . (4)

here; the first term is inter-electronic Coulomb repulsion, the second is the electron-nuclear attraction term, the third is exchange-correlation term, and last term comes from interaction with the external field (in the present case, this interaction is zero). A simple local energy functional form has been applied for the atoms, and the exchange part can be found to be [

δ E x δ ρ = δ E x L D A δ ρ − β [ 4 3 ρ 1 / 3 + 2 3 r 2 ρ α x ( 1 + r 2 ρ 2 / 3 α x ) 2 ] (5)

δ E x L D A δ ρ = − 4 3 C x ρ 1 / 3 . (6)

The simple local parameterized Wigner-type correlation energy functional [

E c = − ∫ ρ a + b ⋅ ρ − 1 / 3 d r (7)

δ E c δ ρ = − a + c ⋅ ρ − 1 / 3 ( a + b ⋅ ρ − 1 / 3 ) 2 (8)

where: a = 9.81 , b = 21.437 , c = 28.582667 are respectively.

The solution of Equation (1) is used split time method, for split time Δ t . It can be written

R ( r , t + Δ t ) ≅ e − Δ t H ^ 0 / 2 e − V ^ Δ t e − 0 Δ t H ^ / 2 R ( r , t ) (9)

One of the main features of the DVR is that a function R ( r , t ) can be approximated by interpolation through the given grid points:

R ( r ) ≅ ∑ j = 0 N R ( r j ) g j ( r ) (10)

here: R ( r j ) is the interpolation function, g j ( r ) is the cardinal function.

The Coulomb wave function is defined by radial grid points. Interpolation function is obtained by using the radial function that is derived from the cardinal functions.

By noting that F ( r ) is the Coulomb function, F ′ ( r ) is the first derivative from F ( r ) at the position r j , ψ j is found to be ψ j = R ( r ) / F ′ ( r ) .

The propagation in the energy space (step first in equation) can now be achieved through

e − H ^ 0 Δ t / 2 R ( r ) = ∑ j = 0 N e − H ^ 0 Δ t / 2 R ( r j ) g j ( r ) . (11)

The cardinal functions g j ( r ) in Equation (10) are given by the following expression

g j ( r ) = 1 F ′ ( r j ) F ( r ) r − r j (12)

where the points are the zeros of the Coulomb wave function F ( r ) and F ′ ( r j ) stands for its first derivative at r j and g j ( r ) satisfies the cardinality condition

g j ( r i ) = δ i j . (13)

Since the Coulomb wave functions was expressed in quadrature rule with expressions for the weight ω j , then DVR basis function F j ( r ) satisfies the eigenvalue for the radial Kohn-Sham type equation:

H ^ ( r ) ψ ( r ) = E ψ ( r ) (14)

and

H ^ ( r ) = − d 2 2 d 2 + V ( r ) . (15)

The DVR greatly simplifies the evaluation of Hamiltonian matrix elements. The potential matrix elements involve merely the evaluation of the interaction potential at the DVR grid points, where no integration is needed.

The DVR basis function f j ( r ) is constructed from the cardinal function g j as follows

f j ( r ) = 1 ω j g j ( r ) , (16)

here the weight ω j is given by [

ω j ≈ π a j 2 . (17)

a j = F ′ ( r j ) (18)

The second derivative of the cardinal function g ″ j is given,

g ″ j ( r j ) = δ j k c k 3 a k − ( 1 − δ j k ) a k a j 2 ( r k − r j ) 2 , (19)

where a k is given by Equation (18) and c k . Here kinetic energy matrix elements D i j calculated using:

c k = − a k ( 2 E + 2 Z / r ) , (20)

D i j = − δ i j c i 6 a i + ( 1 − δ i j ) 1 ( r i − r j ) 2 . (21)

In the Equation (15), to expand R ( r j ) in the eigenvectors of the Hamiltonian H ^ 0 , we first solve the eigenvalue problem for H ^ 0 after discretization of coordinate, the differential equation for this problem can be written as:

∑ j = 1 N [ − 1 2 D i j + V ( r j ) δ i j ] φ k j = ε k φ k j (22)

here D i j denotes the symmetrized second derivative of the cardinal function that is given as,

( D 2 ) i j = 1 3 ( E + Z r ) , i = j (23)

( D 2 ) i j = 1 ( r j − r i ) 2 , i ≠ j (24)

Equation (2) is then numerically solved to achieve a self-consistent set of orbitals, using the DVR method. These orbitals are used to construct various Slater determinants arising out of that particular electronic configuration and its energies computed in the usual manner.

A key step in the time propagation of Equation (9) is to construct the evolution operator e − H ^ l 0 Δ t / 2 ≅ S ( l ) through an accurate and efficient representation of H ^ l 0 . Here we extend the DVR method to achieve optimal grid discretization and an accurate solution of the eigenvalue problem of H ^ l 0 .

In the present work, we are particularly interested in the exploration of the improvement of the Kohn Sham type equation in electron structure calculation. Thus we choose the Slater wave function as our initial state at t = 0 . Note that, the differential equation for time propagation is normalized at each time step. Here the 152 grid points are used for the DVR discretization of the radial coordinates and Δ t = 0.001 a .u , with 500 iterations is used in the time propagation to achieve convergence.

In this section we present results from nonrelativistic electronic structure calculation of the ground states of He, Li, Be, B, C, N and O atoms. Wolfram Mathematica Software has used for the calculations. Here, parameters of the Coulomb wave function such as wave number and effective charges are chosen to be k = 2 E = 3 and Z = 400 .

He | Li | Be | B | C | N | O | ||
---|---|---|---|---|---|---|---|---|

−E | Present work | 2.900 | 7.3197 | 14.582 | 24.779 | 37.9484 | 55.625 | 74.795 |

Hartree-Fock [ | 2.8617 | 7.4332 | 14.573 | 24.529 | 37.688 | 54.400 | 74.809 | |

Roy [ | 2.8973 | 7.221 | 14.221 | 23.964 | 36.953 | 53.407 | 73.451 | |

−Z/r | Present work | 6.853 | 17.054 | 33.447 | 56.728 | 88.447 | 128.915 | 178.317 |

Hartree-Fock [ | 6.749 | 17.115 | 34.072 | 58.143 | 88.649 | 127.326 | 176.324 | |

−Ex | Present work | 1.039 | 1.752 | 2.656 | 3.732 | 5.0416 | 6.527 | 8.223 |

Hartree-Fock [ | 1.026 | 1.781 | 2.667 | 3.744 | 5.045 | 6.596 | 8.174 | |

Roy [ | 1.032 | 1.574 | 2.404 | 3.478 | 4.640 | 5.987 | 7.490 | |

−Ec | Present work | 0.0424 | 0.0659 | 0.093 | 0.1252 | 0.1637 | 0.2058 | 0.2524 |

Hartree-Fock [ | 0.0423 | 0.0435 | 0.094 | 0.111 | 0.1560 | 0.1890 | 0.2412 | |

Roy [ | 0.0423 | 0.154 | 0.322 | 0.302 | 0.368 | 0.434 | 0.543 | |

T | Present work | 2.960 | 7.301 | 14.172 | 23.888 | 37.301 | 53.536 | 74.825 |

Hartree-Fock [ | 2.8617 | 7.433 | 14.573 | 24.529 | 37.688 | 54.401 | 74.810 | |

Roy [ | 2.8974 | 7.382 | 14.844 | 25.300 | 37.924 | 53.664 | 73.444 |

K Roy [

It is satisfying that the CWDVR approach can be used to perform a high precision calculation of the Kohn-Sham type equation with the use of only a few grid points.

Analyses of the results for exchange and correlation energies are given in the same table separately. The results from exchange energies (Ex) calculations of the present calculations show a good agreement with the Hartree-Fock results [

The “exact” correlation energies are considered for the Li, Be, B, C, N and O atoms in

We note that the primary purpose of this work is to explore the feasibility of extending the CWDVR to the solution of the Kohn-Sham type differential equation with imaginary time propagation. LDA-type xc energy functional can be easily adopted in the present CWDVR approach.

The results for total energies are given in the same table separately. The results from total energies calculations of the present calculations show a good agreement with the Hartree-Fock results [

Results from the calculations of radial densities of atoms were created images of correspondent plots. Examples of radial density plots are shown in

Hartree-Fock plot for comparison. Here, the radial density plot shape from our calculation is in good agreement with Hartree-Fock plot [

In this paper, we present that nonrelativistic ground-state properties of He, Li, Be, B, C, N and O atoms can be calculated by means of time-dependent Kohn-Sham equations and an imaginary time evolution methods. The CWDVR approach is shown to be an efficient and precise solution of ground-state energies of atoms. The calculated electronic energies are in good agreement with the Hartree-Fock values and are significantly better than the results in the literature [

This work was supported by the Science Technology Foundation Project (Code: ShUS-2019/08) of Mongolia.

The authors declare no conflicts of interest regarding the publication of this paper.

Davgiikhorol, N., Gonchigsuren, M., Lochin, K., Ochir, S. and Namsrai, T. (2019) Imaginary Time Density Functional Calculation of Ground States of Atoms Using CWDVR Approach. Journal of Modern Physics, 10, 1134-1143. https://doi.org/10.4236/jmp.2019.109073