1

A new method for theoretical calculation of atomic hyperfine structure

Zhao Yukuo1)† Shi Kun2)

1)(School of Mechanical Engineering, Dalian University of Technology, Dalian, 116024,China)

2)(Huazhong University of Science and Technology School of Physics, 430000,China)

Abstract

Schrödinger equation is a nonrelativistic wave equation, which does not have Lorentz invariance. Therefore,

this equation has a large theoretical error in the precise calculation of hydrogen-like system. So the commonly used

method is Dirac-Hartree-Fock approximation in the calculation of atomic system. However, we have found a new

eigen equation, whose eigenvalue of the hydrogen-like system approximates the calculation of quantum

electrodynamics. Hence, we propose a new calculation scheme for the atomic hyperfine structure based on the eigen

equation and the basic principle of Hartree-Fock variational method, and come to our conclusion through the

correlation calculation of excited single states of hydrogen atom, U91+ ion, helium atom and lithium atom as well as

the comparison with NIST, that is, our method is a better improved model of the stationary Schrödinger equation.

Meanwhile, we list the correlation algorithms of energy functional, two-electron coupling integral and radial

generalized integral in the appendix.

Key words: Schrödinger equation; hyperfine structure; magnetic interaction potential; Hartree-Fock method;

variational method;

PACS: 31.15.xt, 31.15.vj, 31.15.A–

E-mail: 229766502@qq.com

1. Introduction

As is known to all, Schrödinger equation is the first principle of quantum mechanics[1-4], which is the calculation

basis of system energy and electron cloud density distribution and has been widely studied by many scholars.

Secondly, Schrödinger equation is a nonrelativistic wave equation, which does not have Lorentz invariance.

Therefore, this equation has a large theoretical error in the precise calculation of hydrogen-like system. So Klein

and Gordon proposed a new relativistic description equation for the single particle motion state in 1926, namely,

Klein-Gordon equation[5,6].

However, Klein-Gordon equation is only applicable to scalar fields (such as π mesons)[7], but not to the

calculation of atomic fine structure, and there are both negative energy and negative probability difficulties.

Therefore, in order to solve this so-called negative probability difficulty, Dirac proposed a new relativistic wave

equation in 1928, namely, Dirac equation[8], and the representation of Dirac equation for the hydrogen-like system

2

is as follows (in atomic unit):

𝐢ħ𝜕

𝜕𝑡φ = (𝐶𝜶 ∙ 𝐏 + �̇�𝐶2 −

𝑍

𝑟)φ (1).

Accordingly, (Dirac energy of hydrogen-like system) can be obtained:

𝐸DH(𝑛𝑖 , 𝑙𝑖) =

1

𝛼2(

𝑛𝑖−𝑙𝑖−1+√(𝑙𝑖+1)2−(𝛼Z)2

√(𝑛𝑖−𝑙𝑖−1)2+(𝑙𝑖+1)

2+2(𝑛𝑖−𝑙𝑖−1)√(𝑙𝑖+1)2−(𝛼Z)2

− 1) (2).

Meanwhile, in other atomic systems, the representation of Dirac equation for the multi-electron system and

the correlation calculation method are shown in Reference [9], the calculation result of this method is often referred

to as the fine structure in the quantum electrodynamics.

In addition, Lamb found 1058MHZ energy level difference between S1/22 and P1/2

2 in 1947[10]. In the

same year, Bethe calculated this according the renormalization theory[11], and the result of low-order approximation

was highly consistent with Lamb’s experimental value[12]. Then, (QED energy of hydrogen-like system) can be

obtained according to his calculation method of gradual development:

𝐸QEDH (𝑛𝑖 , 𝑙𝑖 , 𝑚𝑖 , 𝐽𝑖) = 𝐸D

H(𝑛𝑖 , 𝑙𝑖) + 𝐸LH(𝑛𝑖 , 𝑙𝑖 , 𝐽𝑖) + 𝐸M

H(𝑛𝑖 , 𝑙𝑖 , 𝑚𝑖 , 𝐽𝑖) (3).

Meanwhile, the calculation result of this method is often referred to as the fine structure in the quantum

electrodynamics, as shown in Reference [13].

Wherein, the reduced Planck constant is denoted by ħ, the time is denoted by 𝑡, the light velocity is denoted

by 𝐶, Dirac4 × 4 matrix is denoted by 𝜶 and �̇�, the momentum operator is denoted by 𝐏, the number of nuclear

charges is denoted by 𝑍, the wave function of the single particle is denoted by φ, the principal quantum number is

denoted by 𝑛𝑖 = 1,2⋯, the azimuthal quantum number is denoted by 𝑙𝑖 = 0,1⋯ , (𝑛𝑖 − 1), the magnetic quantum

number is denoted by 𝑚𝑖 = 0,±1⋯ ,±𝑙𝑖, the spin quantum number is denoted by 𝐽𝑖 = 0 𝑜𝑟 1, the fine structure

constant is denoted by 𝛼 ≈1

137.036, and the intermediate function is denoted by

{

𝐸L

H(𝑛, 𝑙, 𝐽) ≈ {4(1−(−1)𝐽)∆2𝑆

H Z4

𝑛3𝑖𝑓(𝑛 > 1 𝑎𝑛𝑑 𝑙 = 0)

0 𝑒𝑙𝑠𝑒

𝐸MH(𝑛, 𝑙,𝑚, 𝐽) ≈ {

3∆1𝑆H ((2𝑙−(−1)𝐽+1)(2𝑙−(−1)𝐽+3)−(2𝑙+1)(2𝑙+3)−3)Z3

8(2𝑙+3)(2𝑙+1)2𝑛3𝑖𝑓(𝑙 = 0 𝑜𝑟 𝑚 ∈ odd number)

3∆1𝑆H ((2𝑙−(−1)𝐽−1)(2𝑙−(−1)𝐽+1)−(2𝑙−1)(2𝑙+1)−3)Z3

8(2𝑙−1)(2𝑙+1)2𝑛3𝑒𝑙𝑠𝑒

(4).

The ground state Lamb shift of hydrogen atoms is denoted by ∆1𝑆H = 0.5556𝛼3, (22S1/2 → 22P1/2) state Lamb

shift is denoted by ∆2𝑆H = 0.4138𝛼3, and the coordinate vector of electron 𝑒𝑖 is denoted by �⃗� 𝑖, as shown in Fig. 1.

3

However, recently, we have discovered a new eigenequation, whose eigenvalue approximates the calculation

result of QED, i.e. (atom):

∑ (𝐻𝑖 ≡ −1

2∇𝑖2 −

𝑍

𝑟𝑖−𝛿3(𝑍)

𝑟𝑖2 −

𝛿2(𝑍)

𝑟𝑖2𝑐𝑜𝑠2(𝜃𝑖)

+𝛿1(𝑍)

𝑟𝑖2𝑠𝑖𝑛2(𝜃𝑖)

+ ∑0.5

𝑟𝑖,𝑗

𝑁𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖 )𝑁

𝑖=1 𝛹 = 𝐸𝛹 (5).

Wherein, the number of extranuclear electrons is denoted by 𝑁, the eigenfunction (wave function) is denoted

by 𝛹, the eigenvalue (system energy) is denoted by 𝐸, the Laplace operator is denoted by

∇𝑖2=

𝜕2

𝜕𝑥𝑖2 +

𝜕2

𝜕𝑦𝑖2 +

𝜕2

𝜕𝑧𝑖2 =

1

𝑟𝑖2

𝜕

𝜕𝑟𝑖(𝑟𝑖

2 𝜕

𝜕𝑟𝑖) +

1

𝑟𝑖2 sin(𝜃𝑖)

𝜕

𝜕𝜃𝑖(sin(𝜃𝑖)

𝜕

𝜕𝜃𝑖) +

1

𝑟𝑖2 sin2(𝜃𝑖)

𝜕2

𝜕𝜙𝑖2,

and a function related to 𝑍 is denoted by 𝛿𝑖(𝑍), as shown in Section 2 below.

Finally, the structure of this paper is as follows: in Section 2, we propose a representation of magnetic potential

and 𝛿𝑖(𝑍) function by analogy of gravitational potential (relativity); in Section 3, we adopt a new trial function

(functional) based on the basic principle of Hartree-Fock variational method[14-17] and propose a new energy

functional minimization model for the atomic system according to the new trial function, and the specific algorithm

is shown in the appendix; the wave equation is a hypothetical theoretical basis in the quantum electrodynamics and

therefore a universal method for theoretical verification compared with the the experimental value, so in Section 4,

we calculated the hyperfine structures of hydrogen atoms, U91+ ions, helium atoms and lithium atoms and

compared with the experimental value of NIST[18] to conclude that Equation (5) and the variational method below

are better calculation schemes for the atomic hyperfine structure.

2. Magnetic potential and 𝜹𝒊(𝒁) function

Suppose that the mass of the stator is denoted by 𝑀𝑠 , the mass of the rotor is denoted by 𝑀𝑟 and the

gravitation constant is denoted by 𝐺 , (the gravitational potential) can be obtained according to Schwarzschild

metric[19]:

Fig. 1: Coordinate vector of electron 𝑒𝑖

�⃗� 𝑖

𝑥

z

𝑦

O

𝑒𝑖

{

𝑟𝑖 = √𝑥𝑖

2 + 𝑦𝑖2 + 𝑧𝑖

2

𝜃𝑖 = 𝐴𝑟𝑐𝑐𝑜𝑠 (𝑧𝑖𝑟𝑖)

𝜙𝑖 = 𝐴𝑟𝑐𝑐𝑜𝑠 (𝑥𝑖

𝑟𝑖sin(𝜃𝑖))

𝑟𝑖,𝑗 = √(𝑥𝑖 − 𝑥𝑗)2+ (𝑦𝑖 − 𝑦𝑗)

2+ (𝑧𝑖 − 𝑧𝑗)

2

𝛽𝑖,𝑗 = 𝐴𝑟𝑐𝑐𝑜𝑠 ቆ𝑟𝑖2 + 𝑟𝑗

2 − 𝑟𝑖,𝑗2

2𝑟𝑖𝑟𝑗ቇ

𝜒𝑖,𝑗 = 𝐴𝑟𝑐𝑐𝑜𝑠 ቆcos(𝜃𝑗) − cos(𝜃𝑖)cos(𝛽𝑖,𝑗)

sin(𝜃𝑖)sin(𝛽𝑖,𝑗)ቇ

�⃗� 𝑗

𝑟𝑖,𝑗 𝑒𝑗

𝛽𝑖,𝑗

𝜒𝑖,𝑗

4

𝑉𝐺(𝑟) ≈ −𝐺𝑀𝑠𝑀𝑟

𝑟−𝐺2𝑀𝑠

2𝑀𝑟

𝐶2𝑟2 (6).

∴Suppose that the carried charge of the stator is denoted by 𝑄𝑠, the carried charge of the rotor is denoted by

𝑄𝑟 , the wlectrostatic force constant is denoted by 𝐾 and a function of 𝑄𝑠 is denoted by 𝛿(𝑄𝑠) , (the

electromagnetic potential) can be obtained by analogy of Equation (6):

𝑉𝐶(𝑟) ≈ −𝐾𝑄𝑠𝑄𝑟

𝑟−𝛿(𝑄𝑠)𝑄𝑟

𝑟2 (7).

So the motion of the charged system around the center generates a 𝑟𝑖−2 related magnetic potential (in the

spherical coordinate system):

𝑉𝐿𝐹(�⃗� 𝑖) = −𝛿3(𝑍)

𝑟𝑖2 −

𝛿2(𝑍)

𝑟𝑖2𝑐𝑜𝑠2(𝜃𝑖)

+𝛿1(𝑍)

𝑟𝑖2𝑠𝑖𝑛2(𝜃𝑖)

(8).

Therefore, the eigen equation we discovered is shown in Equation (5) according to Born-Oppenheimer

approximation[20,21], namely, an improved version of stationary Schrödinger equation.

∴Suppose the stationary eigenequation for the hydrogen-like system to be (improved equation) a ccording to

the variable separation method of Equation (5) and the two-body problem:

{

(

𝑑2

𝑑𝜙2+𝑚𝑖

2)𝛷⟦𝑖⟧(𝜙) = 0

ቆ𝑑2

𝑑𝜃2+cos(𝜃)

sin(𝜃)

𝑑

𝑑𝜃−𝑚𝑖2+2𝛿1(𝑍)

sin2(𝜃)+

2𝛿2(𝑍)

cos2(𝜃)+ 𝐿⟦𝑖⟧(𝐿⟦𝑖⟧ + 1)ቇ𝛩⟦𝑖⟧(𝜃) = 0

(𝑑2

𝑑𝑟2+2

𝑟

𝑑

𝑑𝑟+2𝑍

𝑟−𝐿⟦𝑖⟧(𝐿⟦𝑖⟧+1)−2𝛿3(𝑍)

𝑟2+ 2𝐸⟦𝑖⟧

𝐻 )𝑅⟦𝑖⟧(𝑟) = 0

(9).

Obtain:

{

𝐸⟦𝑖⟧

𝐻 = −1

2𝜉⟦𝑖⟧2 𝑎𝑛𝑑 𝜉⟦𝑖⟧ =

𝑍

𝑛𝑖−𝑙𝑖−1

2+√(𝐿⟦𝑖⟧+

1

2)2−2𝛿3(𝑍)

Φ⟦𝑖⟧(𝜙) = {cos(𝑚𝑖𝜙) 𝑖𝑓(𝑚𝑖 ≥ 0)

sin(|𝑚𝑖|𝜙) 𝑒𝑙𝑠𝑒

𝛩⟦𝑖⟧(𝜃) = 𝑠𝑖𝑛(−1)𝑃𝑖+1√𝑚𝑖

2+2𝛿1(𝑍)(𝜃)∑ 𝑎⟦𝑖⟧,𝑘𝑐𝑜𝑠𝑇⟦𝑖⟧−2𝑘(𝜃)

[𝑙𝑖−|𝑚𝑖|

2]

𝑘=0 (unnormalized)

𝑅⟦𝑖⟧(𝑟) = ∑ 𝑏⟦𝑖⟧,𝑘𝑟𝑘−

1

2+√(𝐿⟦𝑖⟧+

1

2)2−2𝛿3(𝑍)𝑒−𝜉⟦𝑖⟧𝑟

𝑛𝑖−𝑙𝑖−1𝑘=0

(10).

Wherein, the atomic orbital is denoted by ⟦𝑖⟧ = (𝑛𝑖 , 𝑙𝑖 , 𝑚𝑖 , 𝐽𝑖 , 𝑃𝑖; 𝜉⟦𝑖⟧), the parity quantum number is denoted

by 𝑃𝑖 = {0 𝑜𝑟 1 𝑖𝑓(𝑚𝑖 = 0)

1 𝑒𝑙𝑠𝑒 , and the multinomial coefficient is denoted by

{

𝑎⟦𝑖⟧,𝑘 = {

1 𝑖𝑓(𝑘 = 0)

−(𝑇⟦𝑖⟧−2𝑘+2)(𝑇⟦𝑖⟧−2𝑘+1)+2𝛿2(𝑍)

2𝑘(2𝐿⟦𝑖⟧+1−2𝑘)𝑎⟦𝑖⟧,𝑘−1 𝑒𝑙𝑠𝑒

𝑏⟦𝑖⟧,𝑘 = {

1 𝑖𝑓(𝑘 = 0)

−2𝜉⟦𝑖⟧(𝑛𝑖−𝑙𝑖−𝑘)

𝑘ቆ𝑘+√(2𝐿⟦𝑖⟧+1)2−8𝛿3(𝑍)ቇ

𝑏⟦𝑖⟧,𝑘−1 𝑒𝑙𝑠𝑒

(11).

The intermediate function is denoted by

{

𝑇⟦𝑖⟧ = 𝑙𝑖 − |𝑚𝑖| +1

2− (−1)𝐽𝑖√

1

4− 2𝛿2(𝑍)

𝐿⟦𝑖⟧ = 𝑇⟦𝑖⟧ − (−1)𝑃𝑖√𝑚𝑖

2 + 2𝛿1(𝑍)

(12).

5

∴Suppose

{

𝐸�̇�,0,0,0,0

H = −1

𝛼2+

1

𝛼2√1− (

𝛼𝑍

�̇�)2−3∆1𝑆

H 𝑍3

4�̇�3

𝐸�̇�,0,0,0,1H = −

1

𝛼2+

1

𝛼2√1− (

𝛼𝑍

�̇�)2+∆1𝑆H 𝑍3

4�̇�3

𝐸�̇�,0,0,1,0H − 𝐸�̇�+1,0,0,0,0

H =16∆2𝑆

H 𝑍4

(�̇�+1)4

𝑎𝑛𝑑 �̇� = [𝛼𝑍 + 1], can be obtained according to

𝐸⟦𝑖⟧𝐻 = −

1

2𝜉⟦𝑖⟧2 in Equation (10)：

{

𝛿1(𝑍) =

(Λ2(𝑍)−Λ3(𝑍)+16−√(Λ2(𝑍)−Λ3(𝑍)+16)2−64Λ2(𝑍)+64Λ1(𝑍))

2

2048

𝛿2(𝑍) =1

8−(16√2𝛿1(𝑍)−Λ2(𝑍)+Λ1(𝑍))

2

1024𝛿1(𝑍)

𝛿3(𝑍) =1

8(2 − √1 − 8𝛿2(𝑍) − 2√2𝛿1(𝑍))

2−Λ1(𝑍)

8

(13).

Wherein, the intermediate function is denoted by

{

Λ1(𝑍) =

ቆ2𝛼𝑍−(2�̇�−1)√2+1.5∆1𝑆H 𝛼2𝑍3�̇�−3−2√1−(𝛼𝑍)2�̇�−2ቇ

2

2+1.5∆1𝑆H 𝛼2𝑍3�̇�−3−2√1−(𝛼𝑍)2�̇�−2

Λ2(𝑍) =ቆ2𝛼𝑍−(2�̇�−1)√2−0.5∆1𝑆

H 𝛼2𝑍3�̇�−3−2√1−(𝛼𝑍)2�̇�−2ቇ

2

2−0.5∆1𝑆H 𝛼2𝑍3�̇�−3−2√1−(𝛼𝑍)2�̇�−2

Λ3(𝑍) = ((�̇�+1)2(2�̇�+1+√Λ1(𝑍))

√(�̇�+1)4−8∆2𝑆H 𝑍2(2�̇�+1+√Λ1(𝑍))

2− 2�̇� + 1)

2

(14).

3. Variational method

3.1 Trial function

Multi-electron stationary wave equation is a second-order eigenequation without analytical solution, so the

representation of trial function is particularly important in approximate solution (the so-called trial function is the

approximate solution of the eigenfunction in the stationary wave equation).

On the one hand, any single-valued convergent function may become an approximate solution to it

mathematically. On the other hand, the lowest energy is only its partial solution, for example, its approximate

solution does not satisify the lowest energy principle and the orthogonal transformation constraints in the excited

state of the system. In other words, the eigenfunction of the stationary wave equation satisfies this property[22] only

in the case of single electron approximation, for example, Hartree-Fock variational method[14-17], Monte-Carlo

method[23,24] and Kohn-Sham method (or density functional theory)[25] are applied in the calculation of multi-

electron stationary Schrödinger equation. Therefore, the following trial function is adopted according to the basic

principle of Hartree-Fock variational method:

𝛹⟦�⃗⃗� ⟧ = ∑ ∑ (𝜑⟦𝑖⟧(�⃗� 𝑖)𝜑⟦𝑗⟧(�⃗� 𝑗) − (−1)𝑆⟦𝑖⟧,⟦𝑗⟧𝜑⟦𝑖⟧(�⃗� 𝑗)𝜑⟦𝑗⟧(�⃗� 𝑖))∏ 𝜑⟦𝑘⟧(�⃗� 𝑘)

𝑁𝑘≠𝑖,𝑗

𝑁𝑗=𝑖+1

𝑁−1𝑖=1

𝑠. 𝑡. ∀ {𝑆⟦𝑖⟧,⟦𝑗⟧ = 0

|(𝑛𝑖+𝐽𝑖−1)(𝑛𝑖+𝐽𝑖)−(𝑛𝑗+𝐽𝑗−1)(𝑛𝑗+𝐽𝑗)

2+ 𝑙𝑖 + 𝐽𝑖 − 𝑙𝑗 − 𝐽𝑗| < 3

⟹ ⟨𝜑⟦𝑖⟧(�⃗� )|𝜑⟦𝑗⟧(�⃗� )⟩ ≈ 0 (15).

6

Wherein, the hydrogen-like wave function is denoted by 𝜑⟦𝑖⟧(�⃗� ) = 𝐴⟦𝑖⟧𝛷⟦𝑖⟧(𝜙)𝛩⟦𝑖⟧(𝜃)𝑅⟦𝑖⟧(𝑟), the electron

configuration is denoted by ⟦�⃗⃗� ⟧ = (⟦1⟧, ⟦2⟧⋯ ), the normalization coefficient is denoted by

𝐴⟦𝑖⟧ =1

√∫𝛷⟦𝑖⟧2 (𝜙)𝛩⟦𝑖⟧

2 (𝜃)𝑅⟦𝑖⟧2 (𝑟) 𝑑�⃗�

, and the symmetry coefficient sis denoted by 𝑆⟦𝑖⟧,⟦𝑗⟧ = 0 𝑜𝑟 1.

3.2 Energy functional

Suppose that the experimental value of the system is denoted by 𝐸⟦�⃗⃗� ⟧

𝑒𝑥𝑝, the error rate is denoted by

휀⟦�⃗⃗� ⟧ =𝐸⟦�⃗⃗� ⟧−𝐸⟦�⃗⃗� ⟧

𝑒𝑥𝑝

|𝐸⟦�⃗⃗� ⟧

𝑒𝑥𝑝|× 100, and the energy functional minimization model for the helium-like systems is denoted by

𝐸⟦�⃗⃗� ⟧He = 𝑀𝑖𝑛

1

𝐴∑ ⟨𝛹

⟦�⃗⃗� ⟧He|𝐻𝑖|𝛹⟦�⃗⃗� ⟧

He⟩𝑁𝑖=1 (𝑁 = 2 𝑎𝑛𝑑 𝐴 = ⟨𝛹

⟦�⃗⃗� ⟧He |𝛹

⟦�⃗⃗� ⟧He⟩) (16).

Then the maximum error rate of the helium-like system is 𝑀𝑎𝑥 {|휀⟦�⃗⃗� ⟧|} ≈ 0.96 (the specific calculation

process is similar to Hartree Fock method, omitted here).

Wherein, the Hamiltonian operator 𝐻𝑖 is shown in Equation (5), the structure of the trial function 𝛹⟦�⃗⃗� ⟧ is

shown in Equation (15), and the correlation calculation results are shown in Table 1.

Table 1: Energy of helium atom (𝐸⟦�⃗⃗� ⟧S = 𝑀𝑖𝑛

1

𝐴∑ ⟨𝛹

⟦�⃗⃗� ⟧He|−

1

2∇𝑖2 −

2

𝑟𝑖+

0.5

𝑟1,2|𝛹

⟦�⃗⃗� ⟧He⟩2

𝑖=1 𝑎. 𝑢.)

ID 𝑛1, 𝑙1, 𝑚1, 𝐽1 ; 𝑛2, 𝑙2, 𝑚2, 𝐽2; 𝑆⟦1⟧,⟦2⟧ 𝐸⟦�⃗⃗� ⟧He 𝐸

⟦�⃗⃗� ⟧S 𝐸𝐼𝐷

Drake[18,26] 휀⟦�⃗⃗� ⟧

1 1,0,0,0 ; 1,0,0,0 ; 1 -2.875821 -2.875661 -2.90375 0.96

2 1,0,0,0 ; 2,0,0,0 ; 1 -2.170578 -2.170465 -2.17533 0.22

3 1,0,0,0 ; 2,0,0,0 ; 0 -2.138372 -2.138269 -2.14612 0.36

4 1,0,0,0 ; 1,0,0,1 ; 0 -2.130801

5 1,0,0,0 ; 2,1,0,0 ; 0 -2.130799 -2.130691 -2.13332 0.12

6 1,0,0,0 ; 2,1,0,0 ; 1 -2.122499 -2.12239

7 1,0,0,0 ; 3,0,0,0 ; 1 -2.068694 -2.068585 -2.06885 0.01

8 1,0,0,0 ; 3,0,0,0 ; 0 -2.06389 -2.063781 -2.06143 -0.12

9 1,0,0,0 ; 2,0,0,1 ; 0 -2.057419

10 1,0,0,0 ; 3,1,0,0 ; 0 -2.057418 -2.057310 -2.05824 0.04

11 1,0,0,0 ; 3,2,0,0 ; 0 -2.05568 -2.055572 -2.05580 0.01

12 1,0,0,0 ; 3,2,0,0 ; 1 -2.055654 -2.055546 -2.05578 0.01

13 1,0,0,0 ; 3,1,0,0 ; 1 -2.054817 -2.054709

7

In Table 1, the contribution of magnetic interaction (relativistic relativity and quantum electrodynamic

correction) is mainly the correction of kinetic energy and potential energy (the relativistic correction between

electrons is ignored as it is small), and the hyultrafine splitting of the system is much lower than the total energy of

the system. Therefore, the calculation error of Equation (16) mainly arises from the estimation error (repulsive

energy) between electrons caused by the the single particle approximation. So a new monocentric repulsive potential

is introduced under the expression of the trial function of Equation (15):

𝑉𝜃HF(�⃗� 𝑖 , �⃗� 𝑗) =

𝜂1

𝑟𝑖,𝑗+𝜂2𝑟𝑖,𝑗

𝑟𝑖𝑟𝑗+𝜂3(𝑟𝑖

2+𝑟𝑗2)

𝑟𝑖𝑟𝑗𝑟𝑖,𝑗+𝜂4𝑟𝑖,𝑗

2

𝑟𝑖𝑟𝑗2 +

𝜂5(𝑟𝑖+𝑟𝑗)

𝑟𝑖𝑟𝑗𝑟𝑖,𝑗+𝜂6𝑟𝑖𝑟𝑗

𝑟𝑖,𝑗+𝜂7(𝑐𝑜𝑠(𝜃𝑖)𝑐𝑜𝑠(𝜃𝑗)+𝑠𝑖𝑛(𝜃𝑖)𝑠𝑖𝑛(𝜃𝑗)𝑐𝑜𝑠(𝜙𝑖−𝜙𝑗))

𝑟𝑖𝑟𝑗 (17).

Therefore, the Hamiltonian operator in the single particle approximation is (the spherical coordinate system):

𝐻𝑖HF = −

1

2∇𝑖2 −

𝑍

𝑟𝑖−𝛿3(𝑍)

𝑟𝑖2 −

𝛿2(𝑍)

𝑟𝑖2𝑐𝑜𝑠2(𝜃𝑖)

+𝛿1(𝑍)

𝑟𝑖2𝑠𝑖𝑛2(𝜃𝑖)

+ ∑ 𝑉𝜃HF(�⃗� 𝑖, �⃗� 𝑗)

𝑁𝑗=1 𝑎𝑛𝑑 𝑗≠𝑖 (18).

Therefore, the energy functional minimization model of the atomic system is (Rayleigh-Ritz variational method,

and the specific algorithm is shown in the appendix):

𝐸⟦�⃗⃗� ⟧ = 𝑀𝑖𝑛 𝐸𝐼𝐷HF(⟦�⃗⃗� ⟧) =

1

𝐴∑ ⟨𝛹⟦�⃗⃗� ⟧|𝐻𝑖

HF|𝛹⟦�⃗⃗� ⟧⟩𝑁𝑖=1 (19).

Moreover, according to the fitting technology of neural network and 𝑀𝑖𝑛 𝐹(𝜂1, 𝜂2,⋯ 𝜂7) = ∑ |𝐸𝑖−𝐸𝑖

NIST

𝐸𝑖NIST |𝑖=1 ,

it can be obtained(Fitting process, omitted):

𝜂1 = 0.47883387; 𝜂2 = −0.01397390; 𝜂3 = 0.00769582; 𝜂4 = 0.00000713 ;

𝜂5 = 0.00231748; 𝜂6 = 0.01837402; 𝜂7 = −0.1701; (20).

4. Conclusion

4.1 Hyperfine structures of hydrogen atoms and 𝐔𝟗𝟏+ ions (𝒏 ≤ 𝟐)

In order to verify the reasonableness of introducing the magnetic interaction potential 𝑉𝐿𝐹(�⃗� 𝑖), we calculated the

hyrefine structures of hydrogen atoms and U91+ ions, and the calculation results are shown in Table 2 and Table 3.

Table 2: Hyrefine structure of hydrogen atoms (Z=1)

ID 𝑛𝑖 , 𝑙𝑖 , |𝑚𝑖|, 𝐽𝑖 , 𝑃𝑖 Δ𝐸𝑖 Δ𝐸𝑖𝑄𝐸𝐷 휀𝑖

1 1,0,0,0,0 0 0

2 1,0,0,0,1 0.0000002159 0.0000002159 0

3 2,0,0,0,0 0.3750059662 0.3750047181 -0.00033

4 2,0,0,0,1 0.3750059932 0.3750049059 -0.00029

5 1,0,0,1,0 0.3750061270 0.3750063957 0.00007

6 1,0,0,1,1 0.3750061540 0.3750064047 0.00007

7 2,1,0,0,0 0.3750067246 0.3750064002 -0.00009

8

8 2,1,0,0,1 0.3750067381 0.3750064038 -0.00009

9 2,1,1,0,1 0.3750067516

Table 3: Hyrefine structure of U91+ ions (Z=92)

ID 𝑛𝑖 , 𝑙𝑖 , |𝑚𝑖|, 𝐽𝑖 , 𝑃𝑖 Δ𝐸𝑖 Δ𝐸𝑖𝑄𝐸𝐷

휀𝑖

1 1,0,0,0,0 0 0

2 1,0,0,0,1 0.1681208972 0.1681208972

3 2,0,0,0,0 3728.7449184168 3603.9123738739 -3.46

4 2,0,0,0,1 3728.7638243654 3615.4529989734 -3.13

5 1,0,0,1,0 3740.2645284047 3771.7073141378 0.83

6 1,0,0,1,1 3740.2801716006 3771.7143191751 0.83

7 2,1,0,0,0 3799.8628613294 3771.7108166564 -0.75

8 2,1,0,0,1 3799.8700571364 3771.7136186714 -0.75

9 2,1,1,0,1 3799.8772528214

Wherein, Δ𝐸𝑖 = 𝐸⟦�⃗⃗� ⟧ − 𝐸𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑡𝑎𝑡𝑒 , error rate 휀𝑖 =100(Δ𝐸𝑖

𝑄𝐸𝐷−Δ𝐸𝑖)

Δ𝐸𝑖𝑄𝐸𝐷 𝑜𝑟

100(Δ𝐸𝑖NIST−Δ𝐸𝑖)

Δ𝐸𝑖NIST , the calculation

method of 𝐸⟦�⃗⃗� ⟧ is shown in Equation (10) or (19), and the calculation method of 𝐸𝑖𝑄𝐸𝐷

is shown in Equation (3).

Next, in Table 2 and Table 3, Lamb shift= {Δ𝐸2 − Δ𝐸1 = 0.0000002159 𝑜𝑟 0.1681208972Δ𝐸5 − Δ𝐸3 = 0.0000001608 𝑜𝑟 11.5196099879

, which is

consistent with the experimental value(The ground state is shown in Table 6).

However, in Table 2 and Table 3, there are differences in hyperfine structure splitting, which increase with the

increase of Z, because the number of energy levels we calculated is more than the result of quantum electrodynamics,

for example, the number of energy levels we calculated is 7 but the number of energy levels in the quantum

electrodynamics is 6 when 3≤ID≤9. For example, in the hydrogen atoms,

Δ𝐸4−Δ𝐸3

Δ𝐸4𝑄𝐸𝐷

−Δ𝐸3𝑄𝐸𝐷 ≈ 0.14 𝑎𝑛𝑑

Δ𝐸4−Δ𝐸3

𝐸MH(2,0,0,1)−𝐸M

H(2,0,0,0)≈ 1 (21).

Meanwhile, this energy difference does not affect the application of our method in other atomic systems since

this difference is much smaller than the calculation error of electron correlation effect (in the multi-electron system).

In addition, as the difference between two energy levels of hyperfine splitting of hydrogen atoms is not equal

to 1058MHZ (high-order approximation) in the quantum electrodynamics, some unreasonable approximation[12]

9

may exist in the renormalization calculation scheme. So the actual error of the hydrogen-like is less than the

calculation result of Equation (21).

4.2 Hyperfine structures of helium atoms and lithium atoms (excited single state)

Based on the calculation of the hydrogen-like system (as shown in Table 2 and 3), we believe that Equation (5)

is a better improved model of the stationary Schrödinger equation, and it has lower calculation complexity than

Bethe’s calculation method[11].

However, this multi-electron eigenequation has no analytical solution, so a relatively feasible approximate

solution can be obtained only by some approximation, as shown in Equation (19). Therefore, in order to verify the

calculation accuracy of the relevant method, we calculated the excited single state energy of helium atoms and

lithium atoms, and the calculation results are shown in Table 4, Table 5 and Table 6.

Table 4: Excited single state energy of helium atoms (improved), and (𝑛1, 𝑙1, 𝑚1, 𝐽1, 𝑃1) = (1,0,0,0,0).

ID 𝑛2, 𝑙2, |𝑚2|, 𝐽2, 𝑃2; 𝑆⟦1⟧,⟦2⟧ 𝜉1 𝜉2 Δ𝐸𝑖 Δ𝐸𝑖NIST[18]

휀𝑖

1 1,0,0,0,1 ; 1 2.20144 1.20162 0 0

2 2,0,0,0,1 ; 1 2.03659 0.47136 0.7279475 0.7286623 0.10

3 2,0,0,0,0 ; 1 2.03659 0.47136 0.72794754

4 2,0,0,0,0 ; 0 1.87452 0.93726 0.75793283 0.7579329 0.00

5 2,0,0,0,1 ; 0 1.87452 0.93726 0.75793301

6 1,0,0,1,0 ; 0 2.01815 0.53782 0.7704155 0.7707385 0.04

7 1,0,0,1,1 ; 0 2.01815 0.53782 0.77041557 0.7707388 0.04

8 2,1,0,0,0 ; 0 2.01815 0.53780 0.77041825

9 2,1,1,0,1 ; 0 2.01815 0.53780 0.77041829

10 2,1,0,0,1 ; 0 2.01815 0.53780 0.77041832 0.7707434 0.04

11 1,0,0,1,0 ; 1 2.02634 0.51203 0.77443671

12 1,0,0,1,1 ; 1 2.02634 0.51203 0.77443677

13 2,1,0,0,0 ; 1 2.02634 0.51202 0.7744392

14 2,1,1,0,1 ; 1 2.02634 0.51202 0.77443923

15 2,1,0,0,1 ; 1 2.02634 0.51202 0.77443926 0.7800744 0.72

16 3,0,0,0,1 ; 1 2.02724 0.31895 0.83523847 0.8352377 0.00

17 3,0,0,0,0 ; 1 2.02724 0.31895 0.8352385

18 3,0,0,0,0 ; 0 2.02055 0.47273 0.83899804

10

19 3,0,0,0,1 ; 0 2.02055 0.47273 0.83899807 0.8426587 0.43

20 2,0,0,1,0 ; 0 2.02235 0.35736 0.84703949 0.8458483 -0.14

21 2,0,0,1,1 ; 0 2.02235 0.35736 0.84703951 0.8458484 -0.14

22 3,1,0,0,0 ; 0 2.02235 0.35735 0.84704032 0.8458496 -0.14

23 3,1,1,0,1 ; 0 2.02235 0.35735 0.84704033

24 3,1,0,0,1 ; 0 2.02235 0.35735 0.84704034

25 2,1,0,1,1 ; 1 2.02468 0.34700 0.84777506

26 2,1,1,1,1 ; 0 2.02440 0.34723 0.84777856

27 3,2,0,0,0 ; 0 2.02440 0.34722 0.84777931

28 3,2,1,0,1 ; 0 2.02440 0.34722 0.84777932

29 3,2,2,0,1 ; 0 2.02440 0.34722 0.84777932

30 3,2,0,0,1 ; 0 2.02440 0.34722 0.84777933

31 2,1,1,1,1 ; 1 2.02448 0.34667 0.84781357

32 3,2,0,0,0 ; 1 2.02448 0.34666 0.84781432 0.8482960 0.06

33 3,2,1,0,1 ; 1 2.02448 0.34666 0.84781433 0.8482960 0.06

34 3,2,2,0,1 ; 1 2.02448 0.34666 0.84781433

35 3,2,0,0,1 ; 1 2.02448 0.34666 0.84781434 0.8482962 0.06

36 2,1,0,1,1 ; 0 2.02420 0.34689 0.84781711

37 2,0,0,1,0 ; 1 2.02474 0.34660 0.84830709

38 2,0,0,1,1 ; 1 2.02474 0.34660 0.84830711 0.8483116 0.00

39 3,1,0,0,0 ; 1 2.02474 0.34660 0.84830787

40 3,1,1,0,1 ; 1 2.02474 0.34660 0.84830788

41 3,1,0,0,1 ; 1 2.02474 0.34660 0.84830789 0.8487875 0.06

Table 5: Excited single state energy of lithium atoms, and {

(𝑛1, 𝑙1, 𝑚1, 𝐽1, 𝑃1) = (1,0,0,0,0)

(𝑛2, 𝑙2, 𝑚2, 𝐽2, 𝑃2) = (1,0,0,0,1)𝑆⟦1⟧,⟦2⟧ = 1

.

ID 𝑛3, 𝑙3, |𝑚3|, 𝐽3, 𝑃3; 𝑆⟦1⟧,⟦3⟧, 𝑆⟦2⟧,⟦3⟧ 𝜉1 𝜉2 𝜉3 Δ𝐸𝑖 Δ𝐸𝑖NIST[18]

휀𝑖

1 2,0,0,0,1 ; 1, 1 2.7076 2.7112 0.5541 0 0

2 2,0,0,0,0 ; 1, 1 2.7076 2.7112 0.5541 0.00000004

3 1,0,0,1,1 ; 0, 0 2.7091 2.7096 0.5389 0.04221700

11

4 2,1,1,0,1 ; 0, 0 2.7091 2.7096 0.5389 0.04222315

5 2,1,0,0,1 ; 0, 0 2.7091 2.7096 0.5389 0.04222320

6 1,0,0,1,1 ; 1, 0 2.7122 2.7074 0.5366 0.04257838

7 2,1,1,0,1 ; 1, 0 2.7121 2.7074 0.5366 0.04258448

8 2,1,0,0,1 ; 1, 0 2.7122 2.7074 0.5366 0.04258453

9 1,0,0,1,1 ; 1, 1 2.7101 2.7106 0.5333 0.04311297

10 2,1,1,0,1 ; 1, 1 2.7101 2.7106 0.5333 0.04311900

11 2,1,0,0,1 ; 1, 1 2.7101 2.7106 0.5333 0.04311905

12 2,0,0,0,1 ; 1, 0 3.0047 2.1791 1.0895 0.07495433 0.067934 -10.33

13 2,0,0,0,1 ; 0, 0 2.5722 2.5722 1.2861 0.08422833

14 3,0,0,0,1 ; 1, 0 2.7251 2.6973 0.4483 0.08586089

15 3,0,0,0,1 ; 0, 0 2.6923 2.7304 0.5121 0.08936758

16 2,1,0,1,1 ; 1, 0 2.7137 2.7060 0.3724 0.12129450

17 2,1,0,1,1 ; 0, 0 2.7136 2.7060 0.3722 0.12131300

18 2,1,1,1,1 ; 0, 0 2.7101 2.7105 0.3629 0.12253243

19 3,2,1,0,1 ; 0, 0 2.7101 2.7105 0.3629 0.12253430

20 3,2,2,0,1 ; 0, 0 2.7101 2.7105 0.3629 0.12253430

21 3,2,0,0,1 ; 0, 0 2.7101 2.7105 0.3629 0.12253437

22 2,1,1,1,1 ; 1, 0 2.7101 2.7105 0.3628 0.12253596

23 3,2,1,0,1 ; 1, 0 2.7101 2.7105 0.3628 0.12253784

24 3,2,2,0,1 ; 1, 0 2.7101 2.7105 0.3628 0.12253784

25 3,2,0,0,1 ; 1, 0 2.7101 2.7105 0.3628 0.12253790

26 2,1,1,1,1 ; 1, 1 2.7101 2.7105 0.3627 0.12254126

27 3,2,0,0,1 ; 1, 1 2.7101 2.7105 0.3627 0.12254310

28 3,2,1,0,1 ; 1, 1 2.7100 2.7105 0.3627 0.12254313

29 3,2,2,0,1 ; 1, 1 2.7100 2.7105 0.3627 0.12254313

30 2,0,0,1,1 ; 0, 0 2.7094 2.7099 0.3694 0.12293941

31 3,1,1,0,1 ; 0, 0 2.7094 2.7099 0.3693 0.12294147

32 3,1,0,0,1 ; 0, 0 2.7094 2.7099 0.3693 0.12294148

33 2,0,0,1,1 ; 1, 0 2.7105 2.7091 0.3684 0.12306494

12

34 3,1,1,0,1 ; 1, 0 2.7105 2.7091 0.3684 0.12306699

35 3,1,0,0,1 ; 1, 0 2.7105 2.7091 0.3684 0.12306700

36 2,0,0,1,1 ; 1, 1 2.7097 2.7102 0.3670 0.12325151

37 3,1,1,0,1 ; 1, 1 2.7097 2.7102 0.3669 0.12325354

38 3,1,0,0,1 ; 1, 1 2.7097 2.7102 0.3669 0.12325356

39 2,1,0,1,1 ; 1, 1 2.7070 2.7144 0.3544 0.12366534

40 3,0,0,0,1 ; 1, 1 2.7070 2.7084 0.3592 0.12401358 0.124012 0.00

41 3,1,0,1,0 ; 1, 0 2.7124 2.7053 0.3000 0.15232820 0.140965 -8.06

42 3,1,0,1,1 ; 1, 0 2.7124 2.7053 0.3000 0.15232821 0.140965 -8.06

43 3,1,0,1,1 ; 0, 0 2.7123 2.7054 0.2999 0.15234227

44 3,2,1,1,1 ; 0, 0 2.7093 2.7098 0.2871 0.15236303

45 3,2,0,1,1 ; 0, 0 2.7093 2.7098 0.2871 0.15236304

46 3,2,1,1,1 ; 1, 0 2.7093 2.7098 0.2871 0.15236304

47 3,2,2,1,1 ; 1, 0 2.7093 2.7098 0.2871 0.15236304

48 3,2,2,1,1 ; 0, 0 2.7093 2.7098 0.2871 0.15236304

49 3,2,0,1,1 ; 1, 0 2.7093 2.7098 0.2871 0.15236305

50 3,2,2,1,1 ; 1, 1 2.7093 2.7098 0.2871 0.15236305

51 3,2,0,1,1 ; 1, 1 2.7093 2.7098 0.2871 0.15236306

52 3,2,1,1,1 ; 1, 1 2.7093 2.7098 0.2871 0.15236306

53 3,1,1,1,1 ; 0, 0 2.7090 2.7094 0.2902 0.15348358

54 3,1,1,1,1 ; 1, 0 2.7090 2.7094 0.2901 0.15348661

55 3,1,1,1,1 ; 1, 1 2.7090 2.7094 0.2901 0.15349113

56 3,0,0,1,1 ; 0, 0 2.7086 2.7090 0.2951 0.15398605

57 3,0,0,1,0 ; 1, 0 2.7091 2.7086 0.2946 0.15405043 0.142596 -8.03

58 3,0,0,1,1 ; 1, 0 2.7091 2.7086 0.2946 0.15405045 0.142596 -8.03

59 3,0,0,1,1 ; 1, 1 2.7087 2.7092 0.2938 0.15414648

60 3,1,0,1,1 ; 1, 1 2.7062 2.7128 0.2816 0.15452481

Table 6: Ground state energy (ionization energy) of hydrogen atoms, U91+ ions, helium atoms and lithium atoms

Name Z N 𝜉1 𝜉2 𝜉3 𝐸𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑡𝑎𝑡𝑒 𝐸𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑡𝑎𝑡𝑒NIST[18]

휀𝑖

13

H 1 1 1.0000 -0.500007 -0.500007 0.00

U91+ 92 1 98.6035 -4861.323984 -4861.323984 0.00

He 2 2 2.20144 1.20162 -2.90374994 -2.903737 0.00

Li 3 3 2.7076 2.7112 0.5541 -7.47805890 -7.478060 0.00

If the electrons in transition occupy an atomic orbital in the third electron shell in the single electron transition

of helium atom and lithium atom, the number of energy levels of the lithium atom should be more than equal to the

number of energy levels of the helium atom. However, the helium atom has 10 energy levels (1s3s ~ 1s3p(1P°)) in

the calculation results of NIST, and the lithium atom has 5 energy levels (1s23s ~ 1s23d), as shown in Table 4 and

Table 5. Therefore, for the calculation results of the highly excited state of lithium atom, NIST's error may increase,

such as ID =12 in Table 5.

Secondly, the orthogonal calculation is required in the Hartree-Fock method, that is, the wave functions

corresponding to any two eigenvalues satisfy the mutually orthogonal constraints. However, this approximation is

not applicable to the high-precision calculation of the doubly excited state and is not consistent with the fact. For

example, there are always a large number of non-orthogonal cases between two wave functions in the accurate

calculation of the hydrogen molecular ion[27]. If the electron 𝑒2 in the helium-like structure is fixed, the following

equation can be obtained:

(−1

2∇12 −

𝑍

𝑟1+

1

𝑟1,2)𝛹 = (𝐸 +

𝑍

𝑟2)𝛹 (22).

Thus, we can use the method in Reference [27] to obtain the exact solution of Equation (22) (the solving

process is omitted), and there will be a large number of non-orthogonal cases in the calculation results, so the

solution of the wave function does not satisfy the mutually orthogonal constraints in the multi-electron system. In

other words, only when the orbitals occupied by two electrons are far apart, the wave function of the system

approaches the orthogonal transformation, so Drake had such high accuracy in his calculation of helium-like excited

single state system (another reason is that he used a large number of Hylleraas primary functions) [26,28].

Moreover, in the same electron layer, because ∃𝑖 ≠ 𝑗 ⟹ 𝜉𝑖 ≠ 𝜉𝑗 will have lower energy, which is also different

from the method of Drake et al.

Therefore, in order to reduce the complexity of the algorithm, the non-orthogonal method as shown in

Equation (15) is adopted in the construction of the trial function for the multi-electron system, and the calculation

results show that this method is a feasible calculation scheme. For example, in Table 4~6, our error rate |휀𝑖|% is

14

less than 10.33%, which is lower than Grant's calculation method [9]. Thus, our method has good universality in

the comprehensive evaluation of calculation accuracy and complexity, as shown in Equation (5) and Equation (19).

Finally, the trial function is the calculation basis for the electron cloud density distribution and the molecular

structure, and CH4 is a good example to verify the calculation accuracy of the trial function , including the value of

𝑆⟦𝑖⟧,⟦𝑗⟧. Therefore, we will report the theoretical calculation in this aspect in the subsequent research articles.

Appendix I: Energy Functional (Algorithm)

Algorithm name: energy functional 𝐸𝐼𝐷HF(∙);

Input: ⟦�⃗⃗� ⟧; // Configuration ⟦�⃗⃗� ⟧ = (⟦1⟧, ⟦2⟧⋯ ) 𝑎𝑛𝑑 ⟦𝑖⟧ = (𝑛𝑖 , 𝑙𝑖 , 𝑚𝑖 , 𝐽𝑖 , 𝑃𝑖; 𝜉⟦𝑖⟧).

Output: 𝑊 =1

𝐴× ∑ ⟨𝛹⟦�⃗⃗� ⟧|𝐻𝑖

HF|𝛹⟦�⃗⃗� ⟧⟩𝑁𝑖=1 ; // 𝐴 = ⟨𝛹⟦�⃗⃗� ⟧|𝛹⟦�⃗⃗� ⟧⟩.

Algorithmic process:

𝐸𝐼𝐷HF(⟦�⃗⃗� ⟧) {

Initially assigned values:

𝐴 ← 0; 𝑋 ← 0; 𝑊 ← 0;

ℎ1,1 ← 𝜂1; ℎ2,1 ← 𝜂2; ℎ3,1 ← 𝜂3;ℎ4,1 ← 𝜂3; ℎ5,1 ← 𝜂4; ℎ6,1 ← 𝜂5;ℎ7,1 ← 𝜂5; ℎ8,1 ← 𝜂6;

ℎ1,2 ← 0; ℎ2,2 ← −1; ℎ3,2 ← −1;ℎ4,2 ← 1; ℎ5,2 ← −1;ℎ6,2 ← −1;ℎ7,2 ← 0; ℎ8,2 ← 1;

ℎ1,3 ← 0; ℎ2,3 ← −1; ℎ3,3 ← 1;ℎ4,3 ← −1; ℎ5,3 ← −2;ℎ6,3 ← 0;ℎ7,3 ← −1; ℎ8,3 ← 1;

ℎ1,4 ← −1;ℎ2,4 ← 1;ℎ3,4 ← −1;ℎ4,4 ← −1; ℎ5,4 ← 2; ℎ6,4 ← −1;ℎ7,4 ← −1; ℎ8,4 ← −1;

𝐹𝑜𝑟(𝑖 ← 1; 𝑖 ≤ 8; 𝑖 ← 𝑖 + 1){ℎ𝑖,12 ← ℎ𝑖,11 ← ℎ𝑖,10 ← ℎ𝑖,9 ← ℎ𝑖,8 ← ℎ𝑖,7 ← ℎ𝑖,6 ← ℎ𝑖,5 ← 0; }

ℎ9,1 ← 𝜂7; ℎ9,3 ← ℎ9,2 ← −1; ℎ9,12 ← ℎ9,11 ← ℎ9,10 ← ℎ9,9 ← ℎ9,8 ← ℎ9,7 ← ℎ9,4 ← 0; ℎ9,6 ← ℎ9,5 ← 1;

ℎ10,1 ← 𝜂7; ℎ10,3 ← ℎ10,2 ← −1; ℎ10,12 ← ℎ10,11 ← ℎ10,6 ← ℎ10,5 ← ℎ10,4 ← 0;

ℎ11,1 ← 𝜂7; ℎ11,3 ← ℎ11,2 ← −1; ℎ11,10 ← ℎ11,9 ← ℎ11,6 ← ℎ11,5 ← ℎ11,4 ← 0;

ℎ10,10 ← ℎ10,9 ← ℎ10,8 ← ℎ10,7 ← ℎ11,12 ← ℎ11,11 ← ℎ11,8 ← ℎ11,7 ← 1;

𝐹𝑜𝑟(𝑖 ← 1; 𝑖 ≤ 𝑁; 𝑖 ← 𝑖 + 1){ 𝐹𝑜𝑟(𝑗 ← 1; 𝑗 ≤ 𝑁; 𝑗 ← 𝑗 + 1){

𝑆⟦𝑖⟧,⟦𝑗⟧ ← {0 𝑜𝑟 1 𝑖𝑓(⟨𝜑⟦𝑖⟧(�⃗� )|𝜑⟦𝑗⟧(�⃗� )⟩ ≈ 0)

1 𝑒𝑙𝑠𝑒 ；

𝑢𝑖,𝑗 ← ⟨𝜑⟦𝑖⟧|𝜑⟦𝑗⟧⟩; 𝑉𝑖,𝑗 ← ⟨𝜑⟦𝑖⟧|1

𝑟|𝜑⟦𝑗⟧⟩ ; }} //𝜑⟦𝑖⟧ = 𝜑⟦𝑖⟧(�⃗� );

Correlation calculation of potential energy and repulsive energy:

𝐹𝑜𝑟(𝑖1 ← 1; 𝑖1 ≤ 𝑁 − 1; 𝑖1 ← 𝑖1 + 1){ 𝐹𝑜𝑟(𝑗1 ← 𝑖1 + 1; 𝑗1 ≤ 𝑁; 𝑗1 ← 𝑗1 + 1){

𝐹𝑜𝑟(𝑖2 ← 1; 𝑖2 ≤ 𝑁 − 1; 𝑖2 ← 𝑖2 + 1){ 𝐹𝑜𝑟(𝑗2 ← 𝑖2 + 1; 𝑗2 ≤ 𝑁; 𝑗2 ← 𝑗2 + 1){

𝑣1 ← −(−1)𝑆⟦𝑖2⟧,⟦𝑗2⟧;𝑣2 ← −(−1)𝑆⟦𝑖1⟧,⟦𝑗1⟧; 𝑣3 ← (−1)𝑆⟦𝑖1⟧,⟦𝑗1⟧+𝑆⟦𝑖2⟧,⟦𝑗2⟧;

15

Normalization coefficient:

𝐴 ← 𝐴 + 1 + 𝑣1𝑢𝑖2,𝑗22 + 𝑣2𝑢𝑖1,𝑗1

2 ;

𝐼𝑓(𝑖2 = 𝑖1 𝑎𝑛𝑑 𝑗2 = 𝑗1){𝐴 ← 𝐴 + 𝑣3; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖2 = 𝑖1 𝑎𝑛𝑑 𝑗2 ≠ 𝑗1){𝐴 ← 𝐴 + 𝑣3𝑢𝑗1,𝑗2𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖2 = 𝑗1){𝐴 ← 𝐴 + 𝑣3𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2𝑢𝑗1,𝑗2; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗2 = 𝑖1){𝐴 ← 𝐴 + 𝑣3𝑢𝑗1,𝑖2𝑢𝑖1,𝑗1𝑢𝑖2,𝑖1; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗2 = 𝑗1){𝐴 ← 𝐴 + 𝑣3𝑢𝑗1,𝑖1𝑢𝑖1,𝑖2𝑢𝑖2,𝑗1; } 𝐸𝑙𝑠𝑒{𝐴 ← 𝐴 + 𝑣3𝑢𝑖1,𝑗12 𝑢𝑖2,𝑗2

2 ; }

Potential energy:

𝐹𝑜𝑟(𝑖3 ← 1; 𝑖3 ≤ 𝑁; 𝑖3 ← 𝑖3 + 1){

𝑣4 ← 𝜉⟦𝑖3⟧ ቆ𝑛𝑖3 − 𝑙𝑖3 −1

2+√(𝐿⟦𝑖3⟧ +

1

2)2− 2𝛿3(𝑍)ቇ − 𝑍 ; 𝑊 ← 𝑊 + 𝑣4𝑉𝑖3,𝑖3;

𝐼𝑓(𝑖3 = 𝑖2 𝑜𝑟 𝑗2){ 𝑊 ← 𝑊 + 𝑣1𝑣4𝑢𝑖2,𝑗2𝑉𝑖2,𝑗2; } 𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣1𝑣4𝑢𝑖2,𝑗22 𝑉𝑖3,𝑖3; }

𝐼𝑓(𝑖3 = 𝑖1 𝑜𝑟 𝑗1){ 𝑊 ← 𝑊 + 𝑣2𝑣4𝑢𝑖1,𝑗1𝑉𝑖1,𝑗1; } 𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣2𝑣4𝑢𝑖1,𝑗12 𝑉𝑖3,𝑖3; }

𝐼𝑓(𝑖2 = 𝑖1){𝐼𝑓(𝑗2 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑉𝑖3,𝑖3; } 𝐸𝑙𝑠𝑒{𝑖𝑓(𝑖3 = 𝑖1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑗1,𝑗2𝑉𝑖3,𝑗2; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑗1,𝑗2𝑢𝑖1,𝑗2𝑉𝑖3,𝑖1; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2𝑉𝑖3,𝑗1; }

𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2𝑢𝑗1,𝑗2𝑉𝑖3,𝑖3; }}}

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖2 = 𝑗1){𝑖𝑓(𝑖3 = 𝑖1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗2𝑢𝑗1,𝑗2𝑉𝑖3,𝑗1; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2𝑉𝑖3,𝑗2; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑗1,𝑗2𝑉𝑖3,𝑖1; }

𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑗2𝑢𝑗1,𝑗2𝑉𝑖3,𝑖3; }}

𝐸𝑙𝑠𝑒{𝑖𝑓(𝑗2 = 𝑖1){𝑖𝑓(𝑖3 = 𝑖1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑗1,𝑖2𝑉𝑖3,𝑖2; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖2,𝑗1𝑢𝑖1,𝑖2𝑉𝑖3,𝑖1; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑖2𝑉𝑖3,𝑗1; }

𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑖2𝑢𝑗1,𝑖2𝑉𝑖3,𝑖3; }}

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗2 = 𝑗1){ 𝑖𝑓(𝑖3 = 𝑖1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖2,𝑗1𝑢𝑖1,𝑖2𝑉𝑖3,𝑗1; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑖2𝑉𝑖3,𝑖2; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑗1,𝑖2𝑉𝑖3,𝑖1; }

𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖1,𝑖2𝑢𝑗1,𝑖2𝑉𝑖3,𝑖3; }}

𝐸𝑙𝑠𝑒{ 𝑖𝑓(𝑖3 = 𝑖1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖2,𝑗22 𝑉𝑖3,𝑗1; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗1){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1𝑢𝑖2,𝑗22 𝑉𝑖3,𝑖1; } 𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1

2 𝑢𝑗2,𝑖2𝑉𝑖3,𝑗2; }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑗2){𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗12 𝑢𝑗2,𝑖2𝑉𝑖3,𝑖2; }𝐸𝑙𝑠𝑒{𝑊 ← 𝑊 + 𝑣3𝑣4𝑢𝑖1,𝑗1

2 𝑢𝑖2,𝑗22 𝑉𝑖3,𝑖3; }}}}

Repulsive energy between electrons (monocentric double-electron coupling integral is shown in Appendix II below):

𝐹𝑜𝑟(𝑖3 ← 1; 𝑖3 < 𝑁; 𝑖3 ← 𝑖3 + 1){𝐹𝑜𝑟(𝑗3 ← 𝑖3 + 1; 𝑗3 ≤ 𝑁; 𝑗3 ← 𝑗3 + 1){𝐹𝑜𝑟(𝑘 ← 1; 𝑘 ≤ 11; 𝑘 ← 𝑘 + 1){

𝑋 ← 𝑋 + 𝐼II(⟦𝑖3⟧, ⟦𝑖3⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); //�⃗⃗� 𝑘 = (ℎ𝑘,1, ℎ𝑘,2⋯ℎ𝑘,12).

𝐼𝑓(𝑖2 = 𝑖1 𝑎𝑛𝑑 𝑗2 = 𝑗1){𝑋 ← 𝑋 + 𝑣3𝐼II(⟦𝑖3⟧, ⟦𝑖3⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖2 ≠ 𝑖1 𝑎𝑛𝑑 𝑖2 ≠ 𝑗1 𝑎𝑛𝑑 𝑗2 ≠ 𝑗1 𝑎𝑛𝑑 𝑗2 ≠ 𝑖1) {

16

𝐼𝑓(𝑖3 = 𝑖1 𝑎𝑛𝑑 𝑗3 = 𝑗1){𝑋 ← 𝑋 + 𝑣3𝑢𝑖2,𝑗22 𝐼II(⟦𝑖1⟧, ⟦𝑗1⟧, ⟦𝑖1⟧, ⟦𝑗1⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖2 𝑎𝑛𝑑 𝑗3 = 𝑗2){𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗12 𝐼II(⟦𝑖2⟧, ⟦𝑗2⟧, ⟦𝑖2⟧, ⟦𝑗2⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓((𝑖3 = 𝑖1 𝑜𝑟 𝑖3 = 𝑗1) 𝑎𝑛𝑑 (𝑗3 = 𝑖2 𝑜𝑟 𝑗3 = 𝑗2)){𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗1𝑢𝑖2,𝑗2𝐼II(⟦𝑖1⟧, ⟦𝑗1⟧, ⟦𝑖2⟧, ⟦𝑗2⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓((𝑖3 = 𝑖2 𝑜𝑟 𝑖3 = 𝑗2) 𝑎𝑛𝑑 (𝑗3 = 𝑖1 𝑜𝑟 𝑗3 = 𝑗1)){𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗1𝑢𝑖2,𝑗2𝐼II(⟦𝑖1⟧, ⟦𝑗1⟧, ⟦𝑖2⟧, ⟦𝑗2⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖1 𝑜𝑟 𝑖3 = 𝑗1){𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗1𝑢𝑖2,𝑗22 𝐼II(⟦𝑖1⟧, ⟦𝑗1⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑖2 𝑜𝑟 𝑖3 = 𝑗2){𝑋 ← 𝑋 + 𝑣3𝑢𝑖2,𝑗2𝑢𝑖1,𝑗12 𝐼II(⟦𝑖2⟧, ⟦𝑗2⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑖1 𝑜𝑟 𝑗3 = 𝑗1){𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗1𝑢𝑖2,𝑗22 𝐼II(⟦𝑖1⟧, ⟦𝑗1⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑖2 𝑜𝑟 𝑗3 = 𝑗2){𝑋 ← 𝑋 + 𝑣3𝑢𝑖2,𝑗2𝑢𝑖1,𝑗12 𝐼II(⟦𝑖2⟧, ⟦𝑗2⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒{𝑋 ← 𝑋 + 𝑣3𝑢𝑖1,𝑗12 𝑢𝑖2,𝑗2

2 𝐼II(⟦𝑖3⟧, ⟦𝑖3⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

} 𝐹𝑜𝑟(𝑠 ← 1; 𝑠 < 3; 𝑠 ← 𝑠 + 1){

𝐼𝑓(𝑠 = 1){𝑡1 ← 𝑖2; 𝑡2 ← 𝑗2; } 𝐸𝑙𝑠𝑒 {𝑡1 ← 𝑖1; 𝑡2 ← 𝑗1; }

𝐼𝑓(𝑖3 = 𝑡1 𝑎𝑛𝑑 𝑗3 = 𝑡2){𝑋 ← 𝑋 + 𝑣𝑠𝐼II(⟦𝑡1⟧, ⟦𝑡2⟧, ⟦𝑡1⟧, ⟦𝑡2⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑡1 𝑜𝑟 (𝑖3 ≠ 𝑡1 𝑎𝑛𝑑 𝑗3 = 𝑡2)){𝑋 ← 𝑋 + 𝑣𝑠𝑢𝑡1,𝑡2𝐼II(⟦𝑖3⟧, ⟦𝑖3⟧, ⟦𝑡1⟧, ⟦𝑡2⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡2 𝑜𝑟 (𝑖3 = 𝑡1 𝑎𝑛𝑑 𝑗3 ≠ 𝑡2)){𝑋 ← 𝑋 + 𝑣𝑠𝑢𝑡1,𝑡2𝐼II(⟦𝑡1⟧, ⟦𝑡2⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒{𝑋 ← 𝑋 + 𝑣𝑠𝑢𝑡1,𝑡22 𝐼II(⟦𝑖3⟧, ⟦𝑖3⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

} 𝐹𝑜𝑟(𝑠 ← 1; 𝑠 < 7; 𝑠 ← 𝑠 + 1){

𝑖𝑓(𝑠 = 1 ){𝑡1 ← 𝑗2; 𝑡2 ← 𝑖1; 𝑡3 ← 𝑖2; 𝑡4 ← 𝑖1; 𝑡5 ← 𝑗1; 𝑡6 ← 𝑖2; 𝑡7 ← 𝑖1; 𝑡8 ← 𝑖2; 𝑡9 ← 𝑗1; 𝑡10 ← 𝑖1; 𝑡11 ← 𝑗1; }

𝑖𝑓(𝑠 = 2 ){𝑡1 ← 𝑖2; 𝑡2 ← 𝑗1; 𝑡3 ← 𝑖1; 𝑡4 ← 𝑗1; 𝑡5 ← 𝑗2; 𝑡6 ← 𝑖1; 𝑡7 ← 𝑗1; 𝑡8 ← 𝑖1; 𝑡9 ← 𝑗1; 𝑡10 ← 𝑗1; 𝑡11 ← 𝑗2; }

𝑖𝑓(𝑠 = 3 ){𝑡1 ← 𝑖2; 𝑡2 ← 𝑖1; 𝑡3 ← 𝑖1; 𝑡4 ← 𝑗1; 𝑡5 ← 𝑗2; 𝑡6 ← 𝑗1; 𝑡7 ← 𝑗2; 𝑡8 ← 𝑖1; 𝑡9 ← 𝑗1; 𝑡10 ← 𝑖1; 𝑡11 ← 𝑗2; }

𝑖𝑓(𝑠 = 4 ){𝑡1 ← 𝑖2; 𝑡2 ← 𝑖1; 𝑡3 ← 𝑖1; 𝑡4 ← 𝑗2; 𝑡5 ← 𝑗1; 𝑡6 ← 𝑗1; 𝑡7 ← 𝑗2; 𝑡8 ← 𝑖1; 𝑡9 ← 𝑗2; 𝑡10 ← 𝑖1; 𝑡11 ← 𝑗1; }

𝑖𝑓(𝑠 = 5 ){𝑡1 ← 𝑗2; 𝑡2 ← 𝑗1; 𝑡3 ← 𝑖2; 𝑡4 ← 𝑖1; 𝑡5 ← 𝑗1; 𝑡6 ← 𝑖2; 𝑡7 ← 𝑗1; 𝑡8 ← 𝑖1; 𝑡9 ← 𝑗1; 𝑡10 ← 𝑖1; 𝑡11 ← 𝑖2; }

𝑖𝑓(𝑠 = 6 ){𝑡1 ← 𝑗2; 𝑡2 ← 𝑗1; 𝑡3 ← 𝑖1; 𝑡4 ← 𝑖2; 𝑡5 ← 𝑗1; 𝑡6 ← 𝑖1; 𝑡7 ← 𝑗1; 𝑡8 ← 𝑖2; 𝑡9 ← 𝑗1; 𝑡10 ← 𝑖1; 𝑡11 ← 𝑖2; }

𝑖𝑓(𝑡1 = 𝑡2 𝑎𝑛𝑑 𝑡3 < 𝑡4 < 𝑡5){

𝐼𝑓(𝑖3 = 𝑡3 𝑎𝑛𝑑 𝑗3 = 𝑡4){𝑋 ← 𝑋 + 𝑣3𝑢𝑡10,𝑡11𝐼II(⟦𝑡6⟧, ⟦𝑡7⟧, ⟦𝑡8⟧, ⟦𝑡9⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡3 𝑎𝑛𝑑 𝑗3 = 𝑡5){𝑋 ← 𝑋 + 𝑣3𝑢𝑡8,𝑡9𝐼II(⟦𝑡6⟧, ⟦𝑡7⟧, ⟦𝑡10⟧, ⟦𝑡11⟧; �⃗⃗� 𝑘); }

17

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡4 𝑎𝑛𝑑 𝑗3 = 𝑡5){𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝐼II(⟦𝑡8⟧, ⟦𝑡9⟧, ⟦𝑡10⟧, ⟦𝑡11⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡3){𝑋 ← 𝑋 + 𝑣3𝑢𝑡8,𝑡9𝑢𝑡10,𝑡11𝐼II(⟦𝑡6⟧, ⟦𝑡7⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡4){𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝑢𝑡10,𝑡11𝐼II(⟦𝑡8⟧, ⟦𝑡9⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑖3 = 𝑡5){𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝑢𝑡8,𝑡9𝐼II(⟦𝑡10⟧, ⟦𝑡11⟧, ⟦𝑗3⟧, ⟦𝑗3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑡3){𝑋 ← 𝑋 + 𝑣3𝑢𝑡8,𝑡9𝑢𝑡10,𝑡11𝐼II(⟦𝑡6⟧, ⟦𝑡7⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑡4){𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝑢𝑡10,𝑡11𝐼II(⟦𝑡8⟧, ⟦𝑡9⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒 𝑖𝑓(𝑗3 = 𝑡5){𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝑢𝑡8,𝑡9𝐼II(⟦𝑡10⟧, ⟦𝑡11⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

𝐸𝑙𝑠𝑒{𝑋 ← 𝑋 + 𝑣3𝑢𝑡6,𝑡7𝑢𝑡8,𝑡9𝑢𝑡10,𝑡11𝐼II(⟦𝑗3⟧, ⟦𝑗3⟧, ⟦𝑖3⟧, ⟦𝑖3⟧; �⃗⃗� 𝑘); }

}}}}}}}}}

Total energy:

𝑊 ←𝑊+ 2𝑋

𝐴; 𝐹𝑜𝑟(𝑖 ← 1; 𝑖 ≤ 𝑁; 𝑖 ← 𝑖 + 1) {𝑊 ← 𝑊 −

1

2𝜉𝑖2; }

𝑅𝑒𝑡𝑢𝑟𝑛 𝑊; } //End.

Appendix II: Monocentric Double-electron Coupling Integral (Algorithm)

Lemma[26]: six-dimensional integral element 𝑑�⃗� 1𝑑�⃗� 2 = 𝑟1𝑟2𝑟1,2𝑠𝑖𝑛(𝜃1) 𝑑𝑟1𝑑𝑟2𝑑𝑟1,2𝑑𝜙1𝑑𝜃1𝑑𝜒1,2.

Proposition (proof, omitted):

{

cos(𝜃2) = cos(𝜃1)cos(𝛽1,2) + sin(𝜃1)cos(𝜒1,2)sin(𝛽1,2)

cos(𝜙2) =cos(𝛽1,2)cos(𝜙1)

sin(𝜃2)sin(𝜃1)−cos(𝜃2)cos(𝜃1)cos(𝜙1)

sin(𝜃2)sin(𝜃1)+sin(𝛽1,2)sin(𝜒1,2)sin(𝜙1)

sin(𝜃2)

sin(𝜙2) =cos(𝛽1,2)sin(𝜙1)

sin(𝜃2)sin(𝜃1)−cos(𝜃2)cos(𝜃1)sin(𝜙1)

sin(𝜃2)sin(𝜃1)+sin(𝛽1,2)sin(𝜒1,2)cos(𝜙1)

sin(𝜃2)

.

Therefore, the monocentric double-electron coupling integral algorithm we adopted is shown below (the

representation of coordinate vector �⃗� 𝑖 is shown in Figure 1):

Algorithm name: monocentric double-electron coupling integral 𝐼II(∙);

Input: (⟦1⟧, ⟦2⟧, ⟦3⟧, ⟦4⟧; �⃗⃗� );

Output: 𝑊 = ⟨𝜑⟦1⟧(�⃗� 1)𝜑⟦2⟧(�⃗� 1)|ℎ1𝑟1ℎ2𝑟2

ℎ3𝑟1,2ℎ4𝑋(�⃗� 1, �⃗� 2; �⃗⃗� )|𝜑⟦3⟧(�⃗� 2)𝜑⟦4⟧(�⃗� 2)⟩;

// 𝑋(�⃗� 1, �⃗� 2; �⃗⃗� ) = 𝑐𝑜𝑠ℎ5(𝜃1)𝑐𝑜𝑠

ℎ6(𝜃2)𝑠𝑖𝑛ℎ7(𝜃1)𝑠𝑖𝑛

ℎ8(𝜃2)𝑐𝑜𝑠ℎ9(𝜙1)𝑐𝑜𝑠

ℎ10(𝜙2)𝑠𝑖𝑛ℎ11(𝜙1)𝑠𝑖𝑛

ℎ12(𝜙2).

Algorithmic process(The algorithm is not optimized due to length):

𝐼II(⟦1⟧, ⟦2⟧, ⟦3⟧, ⟦4⟧; �⃗⃗� ){ 𝑊 ← 0;

𝐹𝑜𝑟(𝑝1,1 ← 0; 𝑝1,1 < 𝑛1 − 𝑙1; 𝑝1,1 ← 𝑝1,1 + 1){ 𝐹𝑜𝑟 (𝑝1,2 ← 0; 𝑝1,2 ≤ [𝑙1−|𝑚1|

2] ; 𝑝1,2 ← 𝑝1,2 + 1) {

18

𝑞1 ← {0 𝑖𝑓(𝑚1 ≥ 0)

1 𝑒𝑙𝑠𝑒 ; 𝐹𝑜𝑟 (𝑝1,3 ← 0; 𝑝1,3 ≤ [

|𝑚1|−𝑞1

2] ; 𝑝1,3 ← 𝑝1,3 + 1) {

⋮ ⋮

𝐹𝑜𝑟(𝑝4,1 ← 0; 𝑝4,1 < 𝑛4 − 𝑙4; 𝑝4,1 ← 𝑝4,1 + 1){ 𝐹𝑜𝑟 (𝑝4,2 ← 0; 𝑝4,2 ≤ [𝑙4−|𝑚4|

2] ; 𝑝4,2 ← 𝑝4,2 + 1) {

𝑞4 ← {0 𝑖𝑓(𝑚4 ≥ 0)

1 𝑒𝑙𝑠𝑒 ; 𝐹𝑜𝑟 (𝑝4,3 ← 0; 𝑝4,3 ≤ [

|𝑚4|−𝑞4

2] ; 𝑝4,3 ← 𝑝4,3 + 1) {

𝑡1,1 ← 𝑙1 + 𝐽1 − |𝑚1| − 2𝑝1,2 + 𝑙2 + 𝐽2 − |𝑚2| − 2𝑝2,2 + ℎ5; 𝑡1,2 ← |𝑚1| + |𝑚2| + ℎ7;

𝑡2,1 ← 𝑙3 + 𝐽3 − |𝑚3| − 2𝑝3,2 + 𝑙4 + 𝐽4 − |𝑚4| − 2𝑝4,2 + ℎ6; 𝑡2,2 ← |𝑚3| + |𝑚4| + ℎ8;

𝑡1,3 ← |𝑚1| − 𝑞1 − 2𝑝1,3 + |𝑚2| − 𝑞2 − 2𝑝2,3 + ℎ9; 𝑡1,4 ← 𝑞1 + 𝑞2 + ℎ11;

𝑡2,3 ← |𝑚3| − 𝑞3 − 2𝑝3,3 + |𝑚4| − 𝑞4 − 2𝑝4,3 + ℎ10; 𝑡2,4 ← 𝑞3 + 𝑞4 + ℎ12;

𝑡1,5 ← 𝑙1 + 𝐽1 + 𝑝1,1 + 𝑙2 + 𝐽2 + 𝑝2,1 + ℎ2 + 1; 𝑡1,6 ← 𝜉1 + 𝜉2; 𝑡1,7 ← ℎ4 + 1;

𝑡2,5 ← 𝑙3 + 𝐽3 + 𝑝3,1 + 𝑙4 + 𝐽4 + 𝑝4,1 + ℎ3 + 1; 𝑡2,6 ← 𝜉3 + 𝜉4;

𝐼𝑓 (𝑡1,2 < 𝑡2,2) {𝐹𝑜𝑟(𝑖 ← 1; 𝑖 ≤ 6; 𝑖 ← 𝑖 + 1){ 𝑣 ← 𝑡1,𝑖; 𝑡1,𝑖 ← 𝑡2,𝑖; 𝑡2,𝑖 ← 𝑣; }} 𝑡1,2 ← 𝑡1,2 + 1;

𝐼𝑓 (0 ≡ (𝑡2,2 − 𝑡2,4 − 𝑡2,3) 𝑚𝑜𝑑 2){𝑠 ← 0; 𝑘 ← 0; } 𝐸𝑙𝑠𝑒 {𝑠 ← 2; 𝑘 ← 1; }

𝐹𝑜𝑟(𝑖1 ← 0; 𝑖1 ≤ 𝑡2,4; 𝑖1 ← 𝑖1 + 1){ 𝐹𝑜𝑟(𝑖2 ← 0; 𝑖2 ≤ 𝑖1; 𝑖2 ← 𝑖2 + 1){

𝐹𝑜𝑟(𝑖3 ← 0; 𝑖3 ≤ 𝑡2,3; 𝑖3 ← 𝑖3 + 1){ 𝐹𝑜𝑟(𝑖4 ← 0; 𝑖4 ≤ 𝑖3; 𝑖4 ← 𝑖4 + 1){

𝐹𝑜𝑟 (𝑖5 ← 0; 𝑖5 ≤𝑡2,2−𝑡2,4−𝑡2,3−𝑘

2; 𝑖5 ← 𝑖5 + 1) { 𝐹𝑜𝑟(𝑖6 ← 0; 𝑖6 ≤ 𝑠; 𝑖6 ← 𝑖6 + 1){

𝐹𝑜𝑟(𝑖7 ← 0; 𝑖7 ≤ 𝑡2,1 + 𝑖2 + 𝑖4 + 2𝑖5 + 2𝑖6; 𝑖7 ← 𝑖7 + 1){

𝑣 ←(−1)𝑖2+𝑖4+𝑖5𝑡2,3!𝑡2,4!(

𝑡2,2−𝑡2,4−𝑡2,3−𝑘

2)!(2𝑖6)!(𝑡2,1+𝑖2+𝑖4+2𝑖5+2𝑖6)!

4𝑖6(1−2𝑖6)(𝑖6!)2𝑖2!(𝑖1−𝑖2)!(𝑡2,4−𝑖1)!𝑖4!(𝑖3−𝑖4)!(𝑡2,3−𝑖3)!𝑖5!(

𝑡2,2−𝑡2,4−𝑡2,3−𝑘

2−𝑖5)!𝑖7!(𝑡2,1+𝑖2+𝑖4+2𝑖5+2𝑖6−𝑖7)!

;

𝑣 ← 𝑣 × 𝑌I(𝑡1,2 + 𝑖7 − 𝑖1 − 𝑖3, 𝑡1,1 + 𝑡2,1 + 2𝑖2 + 2𝑖4 + 2𝑖5 + 2𝑖6 − 𝑖7);

𝑣 ← 𝑣 × 𝑌II(𝑡2,4 + 𝑡2,3 − 𝑖1 − 𝑖3, 𝑖7) × 𝑌II(𝑡1,4 + 𝑡2,3 + 𝑖1 − 𝑖3, 𝑡1,3 + 𝑡2,4 − 𝑖1 + 𝑖3);

𝑐⟦𝑗⟧,𝑘 ←

{

(−1)𝑘2

𝑚𝑗−2𝑘−1𝑚𝑗(𝑚𝑗−𝑘−1)!

(𝑚𝑗−2𝑘)!𝑘!𝑖𝑓(𝑚𝑗 > 0)

(−1)𝑘2|𝑚𝑗|−2𝑘−1(|𝑚𝑗|−𝑘−1)!

(|𝑚𝑗|−2𝑘−1)!𝑘!𝑒𝑙𝑠𝑒 𝑖𝑓(𝑚𝑗 < 0)

1 𝑒𝑙𝑠𝑒

𝑎𝑛𝑑 (𝑘 = 0,1⋯ , [|𝑚𝑗|−𝑞𝑗

2]) ;

// The expansion coefficient of 𝑐𝑜𝑠(𝑚𝑗𝜙) or 𝑠𝑖𝑛(|𝑚𝑗|𝜙) is denoted by 𝑐⟦𝑗⟧,𝑘;

𝑣 ← 𝑣 ×∏ 𝐴⟦𝑗⟧𝑎⟦𝑗⟧,𝑝𝑗,2𝑏⟦𝑗⟧,𝑝𝑗,1𝑐⟦𝑗⟧,𝑝𝑗,34𝑗=1 ;

𝑊 ←𝑊 + 𝑣 × 𝑌III(𝑡1,7, 𝑡1,5, 𝑡2,5, 𝑡1,6, 𝑡2,6, 𝑡2,4 + 𝑡2,3 − 𝑖1 − 𝑖3 + 𝑖7, 𝑡2,1 + 𝑖1 + 𝑖3 + 2𝑖5 + 2𝑖6 − 𝑖7);

}}}}}}}}}}}}}}}}}}} 𝑅𝑒𝑡𝑢𝑟𝑛 (𝑊 ← 𝑊 × 𝐴1𝐴2𝐴3𝐴4 × ℎ1); } //End.

Appendix III: Radial Generalized Integral (Algorithm)

Algorithm name: radial integral 𝑌III(∙);

19

Input: (�⃗� );

Output: 𝑊 = ∫ 𝑑𝑟1 ∫ 𝑑𝑟2 ∫ 𝑟1,2𝑠1𝑟1

𝑠2𝑟2𝑠3𝑒−(𝑠4𝑟1+𝑠5𝑟2)sin𝑠6(𝛽1,2)cos

𝑠7(𝛽1,2)𝑑𝑟1,2𝑟1+𝑟2|𝑟1−𝑟2|

+∞

0

+∞

0;

Algorithmic process:

𝑌III(�⃗� ){ 𝐼𝑓 (0 ≡ 𝑠6 𝑚𝑜𝑑 2){

𝑊 ← ∑ ∑ ∑(𝑠62)!(𝑠7+2𝑘1)!𝑌V(𝑠4,𝑠5,𝑠2+2𝑘2−2𝑘3−𝑠7−2𝑘1,𝑠3+2𝑘3−𝑠7−2𝑘1,𝑠1+2𝑠7+4𝑘1−2𝑘2)

(−1)𝑠7+𝑘1−𝑘2×2𝑠7+2𝑘1(𝑠62−𝑘1)!𝑘1!(𝑠7+2𝑘1−𝑘2)!(𝑘2−𝑘3)!𝑘3!

𝑘2𝑘3=0

𝑠7+2𝑘1𝑘2=0

𝑠62

𝑘1=0; }

𝐸𝑙𝑠𝑒{

𝑊 ← ∑ ∑ ∑ ∑(2𝑘2)!(

𝑠6−1

2)!(𝑠7+2𝑘1+2𝑘2)!𝑌V(

𝑠4,𝑠5,𝑠2+2𝑘3−2𝑘4−𝑠7−2𝑘1−2𝑘2, 𝑠3+2𝑘4−𝑠7−2𝑘1−2𝑘2,𝑠1+2𝑠7+4𝑘1+4𝑘2−2𝑘3

)

(−1)𝑠7+𝑘1−𝑘3×2𝑠7+2𝑘1+4𝑘2(1−2𝑘2)(𝑘2!)2(𝑠6−1

2−𝑘1)!𝑘1!(𝑠7+2𝑘1+2𝑘2−𝑘3)!(𝑘3−𝑘4)!𝑘4!

𝑘3𝑘4=0

𝑠7+2𝑘1+2𝑘2𝑘3=0

+∞𝑘2=0

𝑠6−1

2

𝑘1=0; }

𝑅𝑒𝑡𝑢𝑟𝑛 𝑊; } //End.

Wherein, the definite integral of the intermediate function is denoted by

{

𝑌I(𝑘, 𝑝) = ∫ sin𝑘(𝑥)cos𝑝(𝑥)

𝜋

0 𝑑𝑥

𝑌II(𝑘, 𝑝) = ∫ sin𝑘(𝑥)cos𝑝(𝑥)𝑑𝑥2𝜋

0

𝑌V(�⃗� ) = ∫ 𝑟1𝑠3𝑒−𝑠1𝑟1𝑑𝑟1 ∫ 𝑟2

𝑠4𝑒−𝑠2𝑟2𝑑𝑟2 ∫ 𝑟1,2𝑠5𝑑𝑟1,2

𝑟1+𝑟2|𝑟1−𝑟2|

+∞

0

+∞

0

.

References:

[1] Schrödinger E 1926 Ann. Phys.79 361.

[2] Schrödinger E 1926 Ann. Phys.79 489.

[3] Schrödinger E 1926 Ann. Phys.80 437.

[4] Schrödinger E 1926 Ann. Phys.81 109.

[5] Klein O 1926 Z.Phys. 37 895.

[6] Gordon W 1926 Z.Phys. 40 117.

[7] Pauli W, Weisskopf V 1934 Helv.Phys.Acta. 7 709.

[8] Dirac P A M 1928 Proc.Roy Soc. 117 610.

[9] Grant I P 1970 Adv. Phys. 19 82.

[10] Lamb W E, Retherford R C 1947 Phys. Rev. 72 214.

[11] Bethe H A 1947 Phys.Rev. 72 339.

[12] Zhu H Y 2013 Quantum field theory (Beijing:Peking University Press)p226 (in Chinese)

[13] Huang S Z 2016 Theory of atomic structure(Beijing: Higher Education Press)p352(in Chinese)

[14] Hartree D R 1928 Proc.Camb.Phil.Soc 24 89

[15] Hartree D R 1933Proc.R.Soc.A 141 209.

20

[16] Fock V Z 1930 Z.Phys. 61 126.

[17] Slater J C 1930 Phys.Rev. 35 210.

[18] NIST Atomic Spectra Database. Energy Levels Data [DB]. National Institute of Standards and

Technology. 2001. 09.09 [2006.05]. http://physics. nist.gov/cgi-bin/AtData/display. Ksh.

[19] Liang C B, Zhou B 2006 Introduction to differential geometry and general relativity(Vol.1)

(Beijing: Science press)p247 (in Chinese)

[20] Born M, Oppenheimer R 1927 Ann. d. Phys. 84 457.

[21] Born M, Huang K 1954 Dynamic Theory of Crystal Lattice(Oxford: Oxford University Preaa).

[22] Xu G X,Li L M,Wang D M 2007 Quantum Chemistry-Basic Principles and ab initio method(Vol.1,2)

(Beijing:Science press)p187,p153 (in Chinese)

[23] Hammond B L, Lester W A Jr, Reynolds P J 1994 Monte Carlo Methods in Ab Ihitio Quantum

Chemistry. (Singapore:World Scientific Publishing Co).

[24] Foulkes W M C, Mitas L, Needs R J, Rajagopal G 2001 Rev.Mod.Phys.73 33.

[25] Kohn W, Sham L J 1965 Phys.Rev.A 140 1133.

[26] Drake G W F, Cassar M M, Nistor R A 2002 Phys.Rev.A. 65 054501.

[27] Scott T C,Aubert-Frécon M, Grotendorst J 2006 Chem.Phys. 324 323.

[28] Hylleraas E A 1928 Z. Phys. 48 469.