1
A new method for theoretical calculation of atomic hyperfine structure
Zhao Yukuo1)โ 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
Schrรถdinger 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 Schrรถdinger equation.
Meanwhile, we list the correlation algorithms of energy functional, two-electron coupling integral and radial
generalized integral in the appendix.
Key words: Schrรถdinger equation; hyperfine structure; magnetic interaction potential; Hartree-Fock method;
variational method;
PACS: 31.15.xt, 31.15.vj, 31.15.Aโ
E-mail: [email protected]
1. Introduction
As is known to all, Schrรถdinger 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, Schrรถdinger 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 ฯ 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):
๐ขฤง๐
๐๐กฯ = (๐ถ๐ถ โ ๐ + ๏ฟฝฬ๏ฟฝ๐ถ2 โ
๐
๐)ฯ (1).
Accordingly, (Dirac energy of hydrogen-like system) can be obtained:
๐ธDH(๐๐ , ๐๐) =
1
๐ผ2(
๐๐โ๐๐โ1+โ(๐๐+1)2โ(๐ผZ)2
โ(๐๐โ๐๐โ1)2+(๐๐+1)
2+2(๐๐โ๐๐โ1)โ(๐๐+1)2โ(๐ผZ)2
โ 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 Lambโs experimental value[12]. Then, (QED energy of hydrogen-like system) can be
obtained according to his calculation method of gradual development:
๐ธQEDH (๐๐ , ๐๐ , ๐๐ , ๐ฝ๐) = ๐ธD
H(๐๐ , ๐๐) + ๐ธLH(๐๐ , ๐๐ , ๐ฝ๐) + ๐ธM
H(๐๐ , ๐๐ , ๐๐ , ๐ฝ๐) (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 ฤง, the time is denoted by ๐ก, the light velocity is denoted
by ๐ถ, Dirac4 ร 4 matrix is denoted by ๐ถ and ๏ฟฝฬ๏ฟฝ, the momentum operator is denoted by ๐, the number of nuclear
charges is denoted by ๐, the wave function of the single particle is denoted by ฯ, the principal quantum number is
denoted by ๐๐ = 1,2โฏ, the azimuthal quantum number is denoted by ๐๐ = 0,1โฏ , (๐๐ โ 1), the magnetic quantum
number is denoted by ๐๐ = 0,ยฑ1โฏ ,ยฑ๐๐, the spin quantum number is denoted by ๐ฝ๐ = 0 ๐๐ 1, the fine structure
constant is denoted by ๐ผ โ1
137.036, and the intermediate function is denoted by
{
๐ธL
H(๐, ๐, ๐ฝ) โ {4(1โ(โ1)๐ฝ)โ2๐
H Z4
๐3๐๐(๐ > 1 ๐๐๐ ๐ = 0)
0 ๐๐๐ ๐
๐ธMH(๐, ๐,๐, ๐ฝ) โ {
3โ1๐H ((2๐โ(โ1)๐ฝ+1)(2๐โ(โ1)๐ฝ+3)โ(2๐+1)(2๐+3)โ3)Z3
8(2๐+3)(2๐+1)2๐3๐๐(๐ = 0 ๐๐ ๐ โ odd number)
3โ1๐H ((2๐โ(โ1)๐ฝโ1)(2๐โ(โ1)๐ฝ+1)โ(2๐โ1)(2๐+1)โ3)Z3
8(2๐โ1)(2๐+1)2๐3๐๐๐ ๐
(4).
The ground state Lamb shift of hydrogen atoms is denoted by โ1๐H = 0.5556๐ผ3, (22S1/2 โ 22P1/2) state Lamb
shift is denoted by โ2๐H = 0.4138๐ผ3, and the coordinate vector of electron ๐๐ is denoted by ๏ฟฝโ๏ฟฝ ๐, 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):
โ (๐ป๐ โก โ1
2โ๐2 โ
๐
๐๐โ๐ฟ3(๐)
๐๐2 โ
๐ฟ2(๐)
๐๐2๐๐๐ 2(๐๐)
+๐ฟ1(๐)
๐๐2๐ ๐๐2(๐๐)
+ โ0.5
๐๐,๐
๐๐=1 ๐๐๐ ๐โ ๐ )๐
๐=1 ๐น = ๐ธ๐น (5).
Wherein, the number of extranuclear electrons is denoted by ๐, the eigenfunction (wave function) is denoted
by ๐น, the eigenvalue (system energy) is denoted by ๐ธ, the Laplace operator is denoted by
โ๐2=
๐2
๐๐ฅ๐2 +
๐2
๐๐ฆ๐2 +
๐2
๐๐ง๐2 =
1
๐๐2
๐
๐๐๐(๐๐
2 ๐
๐๐๐) +
1
๐๐2 sin(๐๐)
๐
๐๐๐(sin(๐๐)
๐
๐๐๐) +
1
๐๐2 sin2(๐๐)
๐2
๐๐๐2,
and a function related to ๐ is denoted by ๐ฟ๐(๐), 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 ๐ฟ๐(๐) 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 ๐น๐(๐) function
Suppose that the mass of the stator is denoted by ๐๐ , the mass of the rotor is denoted by ๐๐ and the
gravitation constant is denoted by ๐บ , (the gravitational potential) can be obtained according to Schwarzschild
metric[19]:
Fig. 1: Coordinate vector of electron ๐๐
๏ฟฝโ๏ฟฝ ๐
๐ฅ
z
๐ฆ
O
๐๐
{
๐๐ = โ๐ฅ๐
2 + ๐ฆ๐2 + ๐ง๐
2
๐๐ = ๐ด๐๐๐๐๐ (๐ง๐๐๐)
๐๐ = ๐ด๐๐๐๐๐ (๐ฅ๐
๐๐sin(๐๐))
๐๐,๐ = โ(๐ฅ๐ โ ๐ฅ๐)2+ (๐ฆ๐ โ ๐ฆ๐)
2+ (๐ง๐ โ ๐ง๐)
2
๐ฝ๐,๐ = ๐ด๐๐๐๐๐ แ๐๐2 + ๐๐
2 โ ๐๐,๐2
2๐๐๐๐แ
๐๐,๐ = ๐ด๐๐๐๐๐ แcos(๐๐) โ cos(๐๐)cos(๐ฝ๐,๐)
sin(๐๐)sin(๐ฝ๐,๐)แ
๏ฟฝโ๏ฟฝ ๐
๐๐,๐ ๐๐
๐ฝ๐,๐
๐๐,๐
4
๐๐บ(๐) โ โ๐บ๐๐ ๐๐
๐โ๐บ2๐๐
2๐๐
๐ถ2๐2 (6).
โดSuppose that the carried charge of the stator is denoted by ๐๐ , the carried charge of the rotor is denoted by
๐๐ , the wlectrostatic force constant is denoted by ๐พ and a function of ๐๐ is denoted by ๐ฟ(๐๐ ) , (the
electromagnetic potential) can be obtained by analogy of Equation (6):
๐๐ถ(๐) โ โ๐พ๐๐ ๐๐
๐โ๐ฟ(๐๐ )๐๐
๐2 (7).
So the motion of the charged system around the center generates a ๐๐โ2 related magnetic potential (in the
spherical coordinate system):
๐๐ฟ๐น(๏ฟฝโ๏ฟฝ ๐) = โ๐ฟ3(๐)
๐๐2 โ
๐ฟ2(๐)
๐๐2๐๐๐ 2(๐๐)
+๐ฟ1(๐)
๐๐2๐ ๐๐2(๐๐)
(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 Schrรถdinger equation.
โดSuppose 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:
{
(
๐2
๐๐2+๐๐
2)๐ทโฆ๐โง(๐) = 0
แ๐2
๐๐2+cos(๐)
sin(๐)
๐
๐๐โ๐๐2+2๐ฟ1(๐)
sin2(๐)+
2๐ฟ2(๐)
cos2(๐)+ ๐ฟโฆ๐โง(๐ฟโฆ๐โง + 1)แ๐ฉโฆ๐โง(๐) = 0
(๐2
๐๐2+2
๐
๐
๐๐+2๐
๐โ๐ฟโฆ๐โง(๐ฟโฆ๐โง+1)โ2๐ฟ3(๐)
๐2+ 2๐ธโฆ๐โง
๐ป )๐ โฆ๐โง(๐) = 0
(9).
Obtain:
{
๐ธโฆ๐โง
๐ป = โ1
2๐โฆ๐โง2 ๐๐๐ ๐โฆ๐โง =
๐
๐๐โ๐๐โ1
2+โ(๐ฟโฆ๐โง+
1
2)2โ2๐ฟ3(๐)
ฮฆโฆ๐โง(๐) = {cos(๐๐๐) ๐๐(๐๐ โฅ 0)
sin(|๐๐|๐) ๐๐๐ ๐
๐ฉโฆ๐โง(๐) = ๐ ๐๐(โ1)๐๐+1โ๐๐
2+2๐ฟ1(๐)(๐)โ ๐โฆ๐โง,๐๐๐๐ ๐โฆ๐โงโ2๐(๐)
[๐๐โ|๐๐|
2]
๐=0 (unnormalized)
๐ โฆ๐โง(๐) = โ ๐โฆ๐โง,๐๐๐โ
1
2+โ(๐ฟโฆ๐โง+
1
2)2โ2๐ฟ3(๐)๐โ๐โฆ๐โง๐
๐๐โ๐๐โ1๐=0
(10).
Wherein, the atomic orbital is denoted by โฆ๐โง = (๐๐ , ๐๐ , ๐๐ , ๐ฝ๐ , ๐๐; ๐โฆ๐โง), the parity quantum number is denoted
by ๐๐ = {0 ๐๐ 1 ๐๐(๐๐ = 0)
1 ๐๐๐ ๐ , and the multinomial coefficient is denoted by
{
๐โฆ๐โง,๐ = {
1 ๐๐(๐ = 0)
โ(๐โฆ๐โงโ2๐+2)(๐โฆ๐โงโ2๐+1)+2๐ฟ2(๐)
2๐(2๐ฟโฆ๐โง+1โ2๐)๐โฆ๐โง,๐โ1 ๐๐๐ ๐
๐โฆ๐โง,๐ = {
1 ๐๐(๐ = 0)
โ2๐โฆ๐โง(๐๐โ๐๐โ๐)
๐แ๐+โ(2๐ฟโฆ๐โง+1)2โ8๐ฟ3(๐)แ
๐โฆ๐โง,๐โ1 ๐๐๐ ๐
(11).
The intermediate function is denoted by
{
๐โฆ๐โง = ๐๐ โ |๐๐| +1
2โ (โ1)๐ฝ๐โ
1
4โ 2๐ฟ2(๐)
๐ฟโฆ๐โง = ๐โฆ๐โง โ (โ1)๐๐โ๐๐
2 + 2๐ฟ1(๐)
(12).
5
โดSuppose
{
๐ธ๏ฟฝฬ๏ฟฝ,0,0,0,0
H = โ1
๐ผ2+
1
๐ผ2โ1โ (
๐ผ๐
๏ฟฝฬ๏ฟฝ)2โ3โ1๐
H ๐3
4๏ฟฝฬ๏ฟฝ3
๐ธ๏ฟฝฬ๏ฟฝ,0,0,0,1H = โ
1
๐ผ2+
1
๐ผ2โ1โ (
๐ผ๐
๏ฟฝฬ๏ฟฝ)2+โ1๐H ๐3
4๏ฟฝฬ๏ฟฝ3
๐ธ๏ฟฝฬ๏ฟฝ,0,0,1,0H โ ๐ธ๏ฟฝฬ๏ฟฝ+1,0,0,0,0
H =16โ2๐
H ๐4
(๏ฟฝฬ๏ฟฝ+1)4
๐๐๐ ๏ฟฝฬ๏ฟฝ = [๐ผ๐ + 1], can be obtained according to
๐ธโฆ๐โง๐ป = โ
1
2๐โฆ๐โง2 in Equation (10)๏ผ
{
๐ฟ1(๐) =
(ฮ2(๐)โฮ3(๐)+16โโ(ฮ2(๐)โฮ3(๐)+16)2โ64ฮ2(๐)+64ฮ1(๐))
2
2048
๐ฟ2(๐) =1
8โ(16โ2๐ฟ1(๐)โฮ2(๐)+ฮ1(๐))
2
1024๐ฟ1(๐)
๐ฟ3(๐) =1
8(2 โ โ1 โ 8๐ฟ2(๐) โ 2โ2๐ฟ1(๐))
2โฮ1(๐)
8
(13).
Wherein, the intermediate function is denoted by
{
ฮ1(๐) =
แ2๐ผ๐โ(2๏ฟฝฬ๏ฟฝโ1)โ2+1.5โ1๐H ๐ผ2๐3๏ฟฝฬ๏ฟฝโ3โ2โ1โ(๐ผ๐)2๏ฟฝฬ๏ฟฝโ2แ
2
2+1.5โ1๐H ๐ผ2๐3๏ฟฝฬ๏ฟฝโ3โ2โ1โ(๐ผ๐)2๏ฟฝฬ๏ฟฝโ2
ฮ2(๐) =แ2๐ผ๐โ(2๏ฟฝฬ๏ฟฝโ1)โ2โ0.5โ1๐
H ๐ผ2๐3๏ฟฝฬ๏ฟฝโ3โ2โ1โ(๐ผ๐)2๏ฟฝฬ๏ฟฝโ2แ
2
2โ0.5โ1๐H ๐ผ2๐3๏ฟฝฬ๏ฟฝโ3โ2โ1โ(๐ผ๐)2๏ฟฝฬ๏ฟฝโ2
ฮ3(๐) = ((๏ฟฝฬ๏ฟฝ+1)2(2๏ฟฝฬ๏ฟฝ+1+โฮ1(๐))
โ(๏ฟฝฬ๏ฟฝ+1)4โ8โ2๐H ๐2(2๏ฟฝฬ๏ฟฝ+1+โฮ1(๐))
2โ 2๏ฟฝฬ๏ฟฝ + 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 Schrรถdinger equation. Therefore, the following trial function is adopted according to the basic
principle of Hartree-Fock variational method:
๐นโฆ๏ฟฝโโ๏ฟฝ โง = โ โ (๐โฆ๐โง(๏ฟฝโ๏ฟฝ ๐)๐โฆ๐โง(๏ฟฝโ๏ฟฝ ๐) โ (โ1)๐โฆ๐โง,โฆ๐โง๐โฆ๐โง(๏ฟฝโ๏ฟฝ ๐)๐โฆ๐โง(๏ฟฝโ๏ฟฝ ๐))โ ๐โฆ๐โง(๏ฟฝโ๏ฟฝ ๐)
๐๐โ ๐,๐
๐๐=๐+1
๐โ1๐=1
๐ . ๐ก. โ {๐โฆ๐โง,โฆ๐โง = 0
|(๐๐+๐ฝ๐โ1)(๐๐+๐ฝ๐)โ(๐๐+๐ฝ๐โ1)(๐๐+๐ฝ๐)
2+ ๐๐ + ๐ฝ๐ โ ๐๐ โ ๐ฝ๐| < 3
โน โจ๐โฆ๐โง(๏ฟฝโ๏ฟฝ )|๐โฆ๐โง(๏ฟฝโ๏ฟฝ )โฉ โ 0 (15).
6
Wherein, the hydrogen-like wave function is denoted by ๐โฆ๐โง(๏ฟฝโ๏ฟฝ ) = ๐ดโฆ๐โง๐ทโฆ๐โง(๐)๐ฉโฆ๐โง(๐)๐ โฆ๐โง(๐), the electron
configuration is denoted by โฆ๏ฟฝโโ๏ฟฝ โง = (โฆ1โง, โฆ2โงโฏ ), the normalization coefficient is denoted by
๐ดโฆ๐โง =1
โโซ๐ทโฆ๐โง2 (๐)๐ฉโฆ๐โง
2 (๐)๐ โฆ๐โง2 (๐) ๐๏ฟฝโ๏ฟฝ
, and the symmetry coefficient sis denoted by ๐โฆ๐โง,โฆ๐โง = 0 ๐๐ 1.
3.2 Energy functional
Suppose that the experimental value of the system is denoted by ๐ธโฆ๏ฟฝโโ๏ฟฝ โง
๐๐ฅ๐, the error rate is denoted by
ํโฆ๏ฟฝโโ๏ฟฝ โง =๐ธโฆ๏ฟฝโโ๏ฟฝ โงโ๐ธโฆ๏ฟฝโโ๏ฟฝ โง
๐๐ฅ๐
|๐ธโฆ๏ฟฝโโ๏ฟฝ โง
๐๐ฅ๐|ร 100, and the energy functional minimization model for the helium-like systems is denoted by
๐ธโฆ๏ฟฝโโ๏ฟฝ โงHe = ๐๐๐
1
๐ดโ โจ๐น
โฆ๏ฟฝโโ๏ฟฝ โงHe|๐ป๐|๐นโฆ๏ฟฝโโ๏ฟฝ โง
Heโฉ๐๐=1 (๐ = 2 ๐๐๐ ๐ด = โจ๐น
โฆ๏ฟฝโโ๏ฟฝ โงHe |๐น
โฆ๏ฟฝโโ๏ฟฝ โงHeโฉ) (16).
Then the maximum error rate of the helium-like system is ๐๐๐ฅ {|ํโฆ๏ฟฝโโ๏ฟฝ โง|} โ 0.96 (the specific calculation
process is similar to Hartree Fock method, omitted here).
Wherein, the Hamiltonian operator ๐ป๐ is shown in Equation (5), the structure of the trial function ๐นโฆ๏ฟฝโโ๏ฟฝ โง is
shown in Equation (15), and the correlation calculation results are shown in Table 1.
Table 1: Energy of helium atom (๐ธโฆ๏ฟฝโโ๏ฟฝ โงS = ๐๐๐
1
๐ดโ โจ๐น
โฆ๏ฟฝโโ๏ฟฝ โงHe|โ
1
2โ๐2 โ
2
๐๐+
0.5
๐1,2|๐น
โฆ๏ฟฝโโ๏ฟฝ โงHeโฉ2
๐=1 ๐. ๐ข.)
ID ๐1, ๐1, ๐1, ๐ฝ1 ; ๐2, ๐2, ๐2, ๐ฝ2; ๐โฆ1โง,โฆ2โง ๐ธโฆ๏ฟฝโโ๏ฟฝ โงHe ๐ธ
โฆ๏ฟฝโโ๏ฟฝ โงS ๐ธ๐ผ๐ท
Drake[18,26] ํโฆ๏ฟฝโโ๏ฟฝ โง
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):
๐๐HF(๏ฟฝโ๏ฟฝ ๐ , ๏ฟฝโ๏ฟฝ ๐) =
๐1
๐๐,๐+๐2๐๐,๐
๐๐๐๐+๐3(๐๐
2+๐๐2)
๐๐๐๐๐๐,๐+๐4๐๐,๐
2
๐๐๐๐2 +
๐5(๐๐+๐๐)
๐๐๐๐๐๐,๐+๐6๐๐๐๐
๐๐,๐+๐7(๐๐๐ (๐๐)๐๐๐ (๐๐)+๐ ๐๐(๐๐)๐ ๐๐(๐๐)๐๐๐ (๐๐โ๐๐))
๐๐๐๐ (17).
Therefore, the Hamiltonian operator in the single particle approximation is (the spherical coordinate system):
๐ป๐HF = โ
1
2โ๐2 โ
๐
๐๐โ๐ฟ3(๐)
๐๐2 โ
๐ฟ2(๐)
๐๐2๐๐๐ 2(๐๐)
+๐ฟ1(๐)
๐๐2๐ ๐๐2(๐๐)
+ โ ๐๐HF(๏ฟฝโ๏ฟฝ ๐, ๏ฟฝโ๏ฟฝ ๐)
๐๐=1 ๐๐๐ ๐โ ๐ (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):
๐ธโฆ๏ฟฝโโ๏ฟฝ โง = ๐๐๐ ๐ธ๐ผ๐ทHF(โฆ๏ฟฝโโ๏ฟฝ โง) =
1
๐ดโ โจ๐นโฆ๏ฟฝโโ๏ฟฝ โง|๐ป๐
HF|๐นโฆ๏ฟฝโโ๏ฟฝ โงโฉ๐๐=1 (19).
Moreover, according to the fitting technology of neural network and ๐๐๐ ๐น(๐1, ๐2,โฏ ๐7) = โ |๐ธ๐โ๐ธ๐
NIST
๐ธ๐NIST |๐=1 ,
it can be obtained(Fitting process, omitted):
๐1 = 0.47883387; ๐2 = โ0.01397390; ๐3 = 0.00769582; ๐4 = 0.00000713 ;
๐5 = 0.00231748; ๐6 = 0.01837402; ๐7 = โ0.1701; (20).
4. Conclusion
4.1 Hyperfine structures of hydrogen atoms and ๐๐๐+ ions (๐ โค ๐)
In order to verify the reasonableness of introducing the magnetic interaction potential ๐๐ฟ๐น(๏ฟฝโ๏ฟฝ ๐), 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 ๐๐ , ๐๐ , |๐๐|, ๐ฝ๐ , ๐๐ ฮ๐ธ๐ ฮ๐ธ๐๐๐ธ๐ท ํ๐
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 ๐๐ , ๐๐ , |๐๐|, ๐ฝ๐ , ๐๐ ฮ๐ธ๐ ฮ๐ธ๐๐๐ธ๐ท
ํ๐
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, ฮ๐ธ๐ = ๐ธโฆ๏ฟฝโโ๏ฟฝ โง โ ๐ธ๐๐๐๐ข๐๐ ๐ ๐ก๐๐ก๐ , error rate ํ๐ =100(ฮ๐ธ๐
๐๐ธ๐ทโฮ๐ธ๐)
ฮ๐ธ๐๐๐ธ๐ท ๐๐
100(ฮ๐ธ๐NISTโฮ๐ธ๐)
ฮ๐ธ๐NIST , the calculation
method of ๐ธโฆ๏ฟฝโโ๏ฟฝ โง is shown in Equation (10) or (19), and the calculation method of ๐ธ๐๐๐ธ๐ท
is shown in Equation (3).
Next, in Table 2 and Table 3, Lamb shift= {ฮ๐ธ2 โ ฮ๐ธ1 = 0.0000002159 ๐๐ 0.1681208972ฮ๐ธ5 โ ฮ๐ธ3 = 0.0000001608 ๐๐ 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 3โคIDโค9. For example, in the hydrogen atoms,
ฮ๐ธ4โฮ๐ธ3
ฮ๐ธ4๐๐ธ๐ท
โฮ๐ธ3๐๐ธ๐ท โ 0.14 ๐๐๐
ฮ๐ธ4โฮ๐ธ3
๐ธMH(2,0,0,1)โ๐ธM
H(2,0,0,0)โ 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 Schrรถdinger equation, and it has lower calculation complexity than
Betheโs 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 (๐1, ๐1, ๐1, ๐ฝ1, ๐1) = (1,0,0,0,0).
ID ๐2, ๐2, |๐2|, ๐ฝ2, ๐2; ๐โฆ1โง,โฆ2โง ๐1 ๐2 ฮ๐ธ๐ ฮ๐ธ๐NIST[18]
ํ๐
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 {
(๐1, ๐1, ๐1, ๐ฝ1, ๐1) = (1,0,0,0,0)
(๐2, ๐2, ๐2, ๐ฝ2, ๐2) = (1,0,0,0,1)๐โฆ1โง,โฆ2โง = 1
.
ID ๐3, ๐3, |๐3|, ๐ฝ3, ๐3; ๐โฆ1โง,โฆ3โง, ๐โฆ2โง,โฆ3โง ๐1 ๐2 ๐3 ฮ๐ธ๐ ฮ๐ธ๐NIST[18]
ํ๐
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 ๐1 ๐2 ๐3 ๐ธ๐๐๐๐ข๐๐ ๐ ๐ก๐๐ก๐ ๐ธ๐๐๐๐ข๐๐ ๐ ๐ก๐๐ก๐NIST[18]
ํ๐
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(1Pยฐ)) 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 ๐2 in the helium-like structure is fixed, the following
equation can be obtained:
(โ1
2โ12 โ
๐
๐1+
1
๐1,2)๐น = (๐ธ +
๐
๐2)๐น (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 โ๐ โ ๐ โน ๐๐ โ ๐๐ 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 |ํ๐|% 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
๐โฆ๐โง,โฆ๐โง. Therefore, we will report the theoretical calculation in this aspect in the subsequent research articles.
Appendix I: Energy Functional (Algorithm)
Algorithm name: energy functional ๐ธ๐ผ๐ทHF(โ);
Input: โฆ๏ฟฝโโ๏ฟฝ โง; // Configuration โฆ๏ฟฝโโ๏ฟฝ โง = (โฆ1โง, โฆ2โงโฏ ) ๐๐๐ โฆ๐โง = (๐๐ , ๐๐ , ๐๐ , ๐ฝ๐ , ๐๐; ๐โฆ๐โง).
Output: ๐ =1
๐ดร โ โจ๐นโฆ๏ฟฝโโ๏ฟฝ โง|๐ป๐
HF|๐นโฆ๏ฟฝโโ๏ฟฝ โงโฉ๐๐=1 ; // ๐ด = โจ๐นโฆ๏ฟฝโโ๏ฟฝ โง|๐นโฆ๏ฟฝโโ๏ฟฝ โงโฉ.
Algorithmic process:
๐ธ๐ผ๐ทHF(โฆ๏ฟฝโโ๏ฟฝ โง) {
Initially assigned values:
๐ด โ 0; ๐ โ 0; ๐ โ 0;
โ1,1 โ ๐1; โ2,1 โ ๐2; โ3,1 โ ๐3;โ4,1 โ ๐3; โ5,1 โ ๐4; โ6,1 โ ๐5;โ7,1 โ ๐5; โ8,1 โ ๐6;
โ1,2 โ 0; โ2,2 โ โ1; โ3,2 โ โ1;โ4,2 โ 1; โ5,2 โ โ1;โ6,2 โ โ1;โ7,2 โ 0; โ8,2 โ 1;
โ1,3 โ 0; โ2,3 โ โ1; โ3,3 โ 1;โ4,3 โ โ1; โ5,3 โ โ2;โ6,3 โ 0;โ7,3 โ โ1; โ8,3 โ 1;
โ1,4 โ โ1;โ2,4 โ 1;โ3,4 โ โ1;โ4,4 โ โ1; โ5,4 โ 2; โ6,4 โ โ1;โ7,4 โ โ1; โ8,4 โ โ1;
๐น๐๐(๐ โ 1; ๐ โค 8; ๐ โ ๐ + 1){โ๐,12 โ โ๐,11 โ โ๐,10 โ โ๐,9 โ โ๐,8 โ โ๐,7 โ โ๐,6 โ โ๐,5 โ 0; }
โ9,1 โ ๐7; โ9,3 โ โ9,2 โ โ1; โ9,12 โ โ9,11 โ โ9,10 โ โ9,9 โ โ9,8 โ โ9,7 โ โ9,4 โ 0; โ9,6 โ โ9,5 โ 1;
โ10,1 โ ๐7; โ10,3 โ โ10,2 โ โ1; โ10,12 โ โ10,11 โ โ10,6 โ โ10,5 โ โ10,4 โ 0;
โ11,1 โ ๐7; โ11,3 โ โ11,2 โ โ1; โ11,10 โ โ11,9 โ โ11,6 โ โ11,5 โ โ11,4 โ 0;
โ10,10 โ โ10,9 โ โ10,8 โ โ10,7 โ โ11,12 โ โ11,11 โ โ11,8 โ โ11,7 โ 1;
๐น๐๐(๐ โ 1; ๐ โค ๐; ๐ โ ๐ + 1){ ๐น๐๐(๐ โ 1; ๐ โค ๐; ๐ โ ๐ + 1){
๐โฆ๐โง,โฆ๐โง โ {0 ๐๐ 1 ๐๐(โจ๐โฆ๐โง(๏ฟฝโ๏ฟฝ )|๐โฆ๐โง(๏ฟฝโ๏ฟฝ )โฉ โ 0)
1 ๐๐๐ ๐ ๏ผ
๐ข๐,๐ โ โจ๐โฆ๐โง|๐โฆ๐โงโฉ; ๐๐,๐ โ โจ๐โฆ๐โง|1
๐|๐โฆ๐โงโฉ ; }} //๐โฆ๐โง = ๐โฆ๐โง(๏ฟฝโ๏ฟฝ );
Correlation calculation of potential energy and repulsive energy:
๐น๐๐(๐1 โ 1; ๐1 โค ๐ โ 1; ๐1 โ ๐1 + 1){ ๐น๐๐(๐1 โ ๐1 + 1; ๐1 โค ๐; ๐1 โ ๐1 + 1){
๐น๐๐(๐2 โ 1; ๐2 โค ๐ โ 1; ๐2 โ ๐2 + 1){ ๐น๐๐(๐2 โ ๐2 + 1; ๐2 โค ๐; ๐2 โ ๐2 + 1){
๐ฃ1 โ โ(โ1)๐โฆ๐2โง,โฆ๐2โง;๐ฃ2 โ โ(โ1)๐โฆ๐1โง,โฆ๐1โง; ๐ฃ3 โ (โ1)๐โฆ๐1โง,โฆ๐1โง+๐โฆ๐2โง,โฆ๐2โง;
15
Normalization coefficient:
๐ด โ ๐ด + 1 + ๐ฃ1๐ข๐2,๐22 + ๐ฃ2๐ข๐1,๐1
2 ;
๐ผ๐(๐2 = ๐1 ๐๐๐ ๐2 = ๐1){๐ด โ ๐ด + ๐ฃ3; } ๐ธ๐๐ ๐ ๐๐(๐2 = ๐1 ๐๐๐ ๐2 โ ๐1){๐ด โ ๐ด + ๐ฃ3๐ข๐1,๐2๐ข๐1,๐1๐ข๐1,๐2; }
๐ธ๐๐ ๐ ๐๐(๐2 = ๐1){๐ด โ ๐ด + ๐ฃ3๐ข๐1,๐1๐ข๐1,๐2๐ข๐1,๐2; } ๐ธ๐๐ ๐ ๐๐(๐2 = ๐1){๐ด โ ๐ด + ๐ฃ3๐ข๐1,๐2๐ข๐1,๐1๐ข๐2,๐1; }
๐ธ๐๐ ๐ ๐๐(๐2 = ๐1){๐ด โ ๐ด + ๐ฃ3๐ข๐1,๐1๐ข๐1,๐2๐ข๐2,๐1; } ๐ธ๐๐ ๐{๐ด โ ๐ด + ๐ฃ3๐ข๐1,๐12 ๐ข๐2,๐2
2 ; }
Potential energy:
๐น๐๐(๐3 โ 1; ๐3 โค ๐; ๐3 โ ๐3 + 1){
๐ฃ4 โ ๐โฆ๐3โง แ๐๐3 โ ๐๐3 โ1
2+โ(๐ฟโฆ๐3โง +
1
2)2โ 2๐ฟ3(๐)แ โ ๐ ; ๐ โ ๐ + ๐ฃ4๐๐3,๐3;
๐ผ๐(๐3 = ๐2 ๐๐ ๐2){ ๐ โ ๐ + ๐ฃ1๐ฃ4๐ข๐2,๐2๐๐2,๐2; } ๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ1๐ฃ4๐ข๐2,๐22 ๐๐3,๐3; }
๐ผ๐(๐3 = ๐1 ๐๐ ๐1){ ๐ โ ๐ + ๐ฃ2๐ฃ4๐ข๐1,๐1๐๐1,๐1; } ๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ2๐ฃ4๐ข๐1,๐12 ๐๐3,๐3; }
๐ผ๐(๐2 = ๐1){๐ผ๐(๐2 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐๐3,๐3; } ๐ธ๐๐ ๐{๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐2; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐1; } ๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐3; }}}
๐ธ๐๐ ๐ ๐๐(๐2 = ๐1){๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐2; } ๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐3; }}
๐ธ๐๐ ๐{๐๐(๐2 = ๐1){๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐2; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐2,๐1๐ข๐1,๐2๐๐3,๐1; } ๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐3; }}
๐ธ๐๐ ๐ ๐๐(๐2 = ๐1){ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐2,๐1๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐2; } ๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐๐3,๐1; }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐1,๐2๐ข๐1,๐2๐๐3,๐3; }}
๐ธ๐๐ ๐{ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐2,๐22 ๐๐3,๐1; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1๐ข๐2,๐22 ๐๐3,๐1; } ๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1
2 ๐ข๐2,๐2๐๐3,๐2; }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐12 ๐ข๐2,๐2๐๐3,๐2; }๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ฃ4๐ข๐1,๐1
2 ๐ข๐2,๐22 ๐๐3,๐3; }}}}
Repulsive energy between electrons (monocentric double-electron coupling integral is shown in Appendix II below):
๐น๐๐(๐3 โ 1; ๐3 < ๐; ๐3 โ ๐3 + 1){๐น๐๐(๐3 โ ๐3 + 1; ๐3 โค ๐; ๐3 โ ๐3 + 1){๐น๐๐(๐ โ 1; ๐ โค 11; ๐ โ ๐ + 1){
๐ โ ๐ + ๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); //๏ฟฝโโ๏ฟฝ ๐ = (โ๐,1, โ๐,2โฏโ๐,12).
๐ผ๐(๐2 = ๐1 ๐๐๐ ๐2 = ๐1){๐ โ ๐ + ๐ฃ3๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐2 โ ๐1 ๐๐๐ ๐2 โ ๐1 ๐๐๐ ๐2 โ ๐1 ๐๐๐ ๐2 โ ๐1) {
16
๐ผ๐(๐3 = ๐1 ๐๐๐ ๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ข๐2,๐22 ๐ผII(โฆ๐1โง, โฆ๐1โง, โฆ๐1โง, โฆ๐1โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐2 ๐๐๐ ๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ข๐1,๐12 ๐ผII(โฆ๐2โง, โฆ๐2โง, โฆ๐2โง, โฆ๐2โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐((๐3 = ๐1 ๐๐ ๐3 = ๐1) ๐๐๐ (๐3 = ๐2 ๐๐ ๐3 = ๐2)){๐ โ ๐ + ๐ฃ3๐ข๐1,๐1๐ข๐2,๐2๐ผII(โฆ๐1โง, โฆ๐1โง, โฆ๐2โง, โฆ๐2โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐((๐3 = ๐2 ๐๐ ๐3 = ๐2) ๐๐๐ (๐3 = ๐1 ๐๐ ๐3 = ๐1)){๐ โ ๐ + ๐ฃ3๐ข๐1,๐1๐ข๐2,๐2๐ผII(โฆ๐1โง, โฆ๐1โง, โฆ๐2โง, โฆ๐2โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1 ๐๐ ๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ข๐1,๐1๐ข๐2,๐22 ๐ผII(โฆ๐1โง, โฆ๐1โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐2 ๐๐ ๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ข๐2,๐2๐ข๐1,๐12 ๐ผII(โฆ๐2โง, โฆ๐2โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐1 ๐๐ ๐3 = ๐1){๐ โ ๐ + ๐ฃ3๐ข๐1,๐1๐ข๐2,๐22 ๐ผII(โฆ๐1โง, โฆ๐1โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐2 ๐๐ ๐3 = ๐2){๐ โ ๐ + ๐ฃ3๐ข๐2,๐2๐ข๐1,๐12 ๐ผII(โฆ๐2โง, โฆ๐2โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ข๐1,๐12 ๐ข๐2,๐2
2 ๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
} ๐น๐๐(๐ โ 1; ๐ < 3; ๐ โ ๐ + 1){
๐ผ๐(๐ = 1){๐ก1 โ ๐2; ๐ก2 โ ๐2; } ๐ธ๐๐ ๐ {๐ก1 โ ๐1; ๐ก2 โ ๐1; }
๐ผ๐(๐3 = ๐ก1 ๐๐๐ ๐3 = ๐ก2){๐ โ ๐ + ๐ฃ๐ ๐ผII(โฆ๐ก1โง, โฆ๐ก2โง, โฆ๐ก1โง, โฆ๐ก2โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก1 ๐๐ (๐3 โ ๐ก1 ๐๐๐ ๐3 = ๐ก2)){๐ โ ๐ + ๐ฃ๐ ๐ข๐ก1,๐ก2๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐ก1โง, โฆ๐ก2โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก2 ๐๐ (๐3 = ๐ก1 ๐๐๐ ๐3 โ ๐ก2)){๐ โ ๐ + ๐ฃ๐ ๐ข๐ก1,๐ก2๐ผII(โฆ๐ก1โง, โฆ๐ก2โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ๐ ๐ข๐ก1,๐ก22 ๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
} ๐น๐๐(๐ โ 1; ๐ < 7; ๐ โ ๐ + 1){
๐๐(๐ = 1 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐2; ๐ก4 โ ๐1; ๐ก5 โ ๐1; ๐ก6 โ ๐2; ๐ก7 โ ๐1; ๐ก8 โ ๐2; ๐ก9 โ ๐1; ๐ก10 โ ๐1; ๐ก11 โ ๐1; }
๐๐(๐ = 2 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐1; ๐ก4 โ ๐1; ๐ก5 โ ๐2; ๐ก6 โ ๐1; ๐ก7 โ ๐1; ๐ก8 โ ๐1; ๐ก9 โ ๐1; ๐ก10 โ ๐1; ๐ก11 โ ๐2; }
๐๐(๐ = 3 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐1; ๐ก4 โ ๐1; ๐ก5 โ ๐2; ๐ก6 โ ๐1; ๐ก7 โ ๐2; ๐ก8 โ ๐1; ๐ก9 โ ๐1; ๐ก10 โ ๐1; ๐ก11 โ ๐2; }
๐๐(๐ = 4 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐1; ๐ก4 โ ๐2; ๐ก5 โ ๐1; ๐ก6 โ ๐1; ๐ก7 โ ๐2; ๐ก8 โ ๐1; ๐ก9 โ ๐2; ๐ก10 โ ๐1; ๐ก11 โ ๐1; }
๐๐(๐ = 5 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐2; ๐ก4 โ ๐1; ๐ก5 โ ๐1; ๐ก6 โ ๐2; ๐ก7 โ ๐1; ๐ก8 โ ๐1; ๐ก9 โ ๐1; ๐ก10 โ ๐1; ๐ก11 โ ๐2; }
๐๐(๐ = 6 ){๐ก1 โ ๐2; ๐ก2 โ ๐1; ๐ก3 โ ๐1; ๐ก4 โ ๐2; ๐ก5 โ ๐1; ๐ก6 โ ๐1; ๐ก7 โ ๐1; ๐ก8 โ ๐2; ๐ก9 โ ๐1; ๐ก10 โ ๐1; ๐ก11 โ ๐2; }
๐๐(๐ก1 = ๐ก2 ๐๐๐ ๐ก3 < ๐ก4 < ๐ก5){
๐ผ๐(๐3 = ๐ก3 ๐๐๐ ๐3 = ๐ก4){๐ โ ๐ + ๐ฃ3๐ข๐ก10,๐ก11๐ผII(โฆ๐ก6โง, โฆ๐ก7โง, โฆ๐ก8โง, โฆ๐ก9โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก3 ๐๐๐ ๐3 = ๐ก5){๐ โ ๐ + ๐ฃ3๐ข๐ก8,๐ก9๐ผII(โฆ๐ก6โง, โฆ๐ก7โง, โฆ๐ก10โง, โฆ๐ก11โง; ๏ฟฝโโ๏ฟฝ ๐); }
17
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก4 ๐๐๐ ๐3 = ๐ก5){๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ผII(โฆ๐ก8โง, โฆ๐ก9โง, โฆ๐ก10โง, โฆ๐ก11โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก3){๐ โ ๐ + ๐ฃ3๐ข๐ก8,๐ก9๐ข๐ก10,๐ก11๐ผII(โฆ๐ก6โง, โฆ๐ก7โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก4){๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ข๐ก10,๐ก11๐ผII(โฆ๐ก8โง, โฆ๐ก9โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก5){๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ข๐ก8,๐ก9๐ผII(โฆ๐ก10โง, โฆ๐ก11โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก3){๐ โ ๐ + ๐ฃ3๐ข๐ก8,๐ก9๐ข๐ก10,๐ก11๐ผII(โฆ๐ก6โง, โฆ๐ก7โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก4){๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ข๐ก10,๐ก11๐ผII(โฆ๐ก8โง, โฆ๐ก9โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐ ๐๐(๐3 = ๐ก5){๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ข๐ก8,๐ก9๐ผII(โฆ๐ก10โง, โฆ๐ก11โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
๐ธ๐๐ ๐{๐ โ ๐ + ๐ฃ3๐ข๐ก6,๐ก7๐ข๐ก8,๐ก9๐ข๐ก10,๐ก11๐ผII(โฆ๐3โง, โฆ๐3โง, โฆ๐3โง, โฆ๐3โง; ๏ฟฝโโ๏ฟฝ ๐); }
}}}}}}}}}
Total energy:
๐ โ๐+ 2๐
๐ด; ๐น๐๐(๐ โ 1; ๐ โค ๐; ๐ โ ๐ + 1) {๐ โ ๐ โ
1
2๐๐2; }
๐ ๐๐ก๐ข๐๐ ๐; } //End.
Appendix II: Monocentric Double-electron Coupling Integral (Algorithm)
Lemma[26]: six-dimensional integral element ๐๏ฟฝโ๏ฟฝ 1๐๏ฟฝโ๏ฟฝ 2 = ๐1๐2๐1,2๐ ๐๐(๐1) ๐๐1๐๐2๐๐1,2๐๐1๐๐1๐๐1,2.
Proposition (proof, omitted):
{
cos(๐2) = cos(๐1)cos(๐ฝ1,2) + sin(๐1)cos(๐1,2)sin(๐ฝ1,2)
cos(๐2) =cos(๐ฝ1,2)cos(๐1)
sin(๐2)sin(๐1)โcos(๐2)cos(๐1)cos(๐1)
sin(๐2)sin(๐1)+sin(๐ฝ1,2)sin(๐1,2)sin(๐1)
sin(๐2)
sin(๐2) =cos(๐ฝ1,2)sin(๐1)
sin(๐2)sin(๐1)โcos(๐2)cos(๐1)sin(๐1)
sin(๐2)sin(๐1)+sin(๐ฝ1,2)sin(๐1,2)cos(๐1)
sin(๐2)
.
Therefore, the monocentric double-electron coupling integral algorithm we adopted is shown below (the
representation of coordinate vector ๏ฟฝโ๏ฟฝ ๐ is shown in Figure 1):
Algorithm name: monocentric double-electron coupling integral ๐ผII(โ);
Input: (โฆ1โง, โฆ2โง, โฆ3โง, โฆ4โง; ๏ฟฝโโ๏ฟฝ );
Output: ๐ = โจ๐โฆ1โง(๏ฟฝโ๏ฟฝ 1)๐โฆ2โง(๏ฟฝโ๏ฟฝ 1)|โ1๐1โ2๐2
โ3๐1,2โ4๐(๏ฟฝโ๏ฟฝ 1, ๏ฟฝโ๏ฟฝ 2; ๏ฟฝโโ๏ฟฝ )|๐โฆ3โง(๏ฟฝโ๏ฟฝ 2)๐โฆ4โง(๏ฟฝโ๏ฟฝ 2)โฉ;
// ๐(๏ฟฝโ๏ฟฝ 1, ๏ฟฝโ๏ฟฝ 2; ๏ฟฝโโ๏ฟฝ ) = ๐๐๐ โ5(๐1)๐๐๐
โ6(๐2)๐ ๐๐โ7(๐1)๐ ๐๐
โ8(๐2)๐๐๐ โ9(๐1)๐๐๐
โ10(๐2)๐ ๐๐โ11(๐1)๐ ๐๐
โ12(๐2).
Algorithmic process(The algorithm is not optimized due to length):
๐ผII(โฆ1โง, โฆ2โง, โฆ3โง, โฆ4โง; ๏ฟฝโโ๏ฟฝ ){ ๐ โ 0;
๐น๐๐(๐1,1 โ 0; ๐1,1 < ๐1 โ ๐1; ๐1,1 โ ๐1,1 + 1){ ๐น๐๐ (๐1,2 โ 0; ๐1,2 โค [๐1โ|๐1|
2] ; ๐1,2 โ ๐1,2 + 1) {
18
๐1 โ {0 ๐๐(๐1 โฅ 0)
1 ๐๐๐ ๐ ; ๐น๐๐ (๐1,3 โ 0; ๐1,3 โค [
|๐1|โ๐1
2] ; ๐1,3 โ ๐1,3 + 1) {
โฎ โฎ
๐น๐๐(๐4,1 โ 0; ๐4,1 < ๐4 โ ๐4; ๐4,1 โ ๐4,1 + 1){ ๐น๐๐ (๐4,2 โ 0; ๐4,2 โค [๐4โ|๐4|
2] ; ๐4,2 โ ๐4,2 + 1) {
๐4 โ {0 ๐๐(๐4 โฅ 0)
1 ๐๐๐ ๐ ; ๐น๐๐ (๐4,3 โ 0; ๐4,3 โค [
|๐4|โ๐4
2] ; ๐4,3 โ ๐4,3 + 1) {
๐ก1,1 โ ๐1 + ๐ฝ1 โ |๐1| โ 2๐1,2 + ๐2 + ๐ฝ2 โ |๐2| โ 2๐2,2 + โ5; ๐ก1,2 โ |๐1| + |๐2| + โ7;
๐ก2,1 โ ๐3 + ๐ฝ3 โ |๐3| โ 2๐3,2 + ๐4 + ๐ฝ4 โ |๐4| โ 2๐4,2 + โ6; ๐ก2,2 โ |๐3| + |๐4| + โ8;
๐ก1,3 โ |๐1| โ ๐1 โ 2๐1,3 + |๐2| โ ๐2 โ 2๐2,3 + โ9; ๐ก1,4 โ ๐1 + ๐2 + โ11;
๐ก2,3 โ |๐3| โ ๐3 โ 2๐3,3 + |๐4| โ ๐4 โ 2๐4,3 + โ10; ๐ก2,4 โ ๐3 + ๐4 + โ12;
๐ก1,5 โ ๐1 + ๐ฝ1 + ๐1,1 + ๐2 + ๐ฝ2 + ๐2,1 + โ2 + 1; ๐ก1,6 โ ๐1 + ๐2; ๐ก1,7 โ โ4 + 1;
๐ก2,5 โ ๐3 + ๐ฝ3 + ๐3,1 + ๐4 + ๐ฝ4 + ๐4,1 + โ3 + 1; ๐ก2,6 โ ๐3 + ๐4;
๐ผ๐ (๐ก1,2 < ๐ก2,2) {๐น๐๐(๐ โ 1; ๐ โค 6; ๐ โ ๐ + 1){ ๐ฃ โ ๐ก1,๐; ๐ก1,๐ โ ๐ก2,๐; ๐ก2,๐ โ ๐ฃ; }} ๐ก1,2 โ ๐ก1,2 + 1;
๐ผ๐ (0 โก (๐ก2,2 โ ๐ก2,4 โ ๐ก2,3) ๐๐๐ 2){๐ โ 0; ๐ โ 0; } ๐ธ๐๐ ๐ {๐ โ 2; ๐ โ 1; }
๐น๐๐(๐1 โ 0; ๐1 โค ๐ก2,4; ๐1 โ ๐1 + 1){ ๐น๐๐(๐2 โ 0; ๐2 โค ๐1; ๐2 โ ๐2 + 1){
๐น๐๐(๐3 โ 0; ๐3 โค ๐ก2,3; ๐3 โ ๐3 + 1){ ๐น๐๐(๐4 โ 0; ๐4 โค ๐3; ๐4 โ ๐4 + 1){
๐น๐๐ (๐5 โ 0; ๐5 โค๐ก2,2โ๐ก2,4โ๐ก2,3โ๐
2; ๐5 โ ๐5 + 1) { ๐น๐๐(๐6 โ 0; ๐6 โค ๐ ; ๐6 โ ๐6 + 1){
๐น๐๐(๐7 โ 0; ๐7 โค ๐ก2,1 + ๐2 + ๐4 + 2๐5 + 2๐6; ๐7 โ ๐7 + 1){
๐ฃ โ(โ1)๐2+๐4+๐5๐ก2,3!๐ก2,4!(
๐ก2,2โ๐ก2,4โ๐ก2,3โ๐
2)!(2๐6)!(๐ก2,1+๐2+๐4+2๐5+2๐6)!
4๐6(1โ2๐6)(๐6!)2๐2!(๐1โ๐2)!(๐ก2,4โ๐1)!๐4!(๐3โ๐4)!(๐ก2,3โ๐3)!๐5!(
๐ก2,2โ๐ก2,4โ๐ก2,3โ๐
2โ๐5)!๐7!(๐ก2,1+๐2+๐4+2๐5+2๐6โ๐7)!
;
๐ฃ โ ๐ฃ ร ๐I(๐ก1,2 + ๐7 โ ๐1 โ ๐3, ๐ก1,1 + ๐ก2,1 + 2๐2 + 2๐4 + 2๐5 + 2๐6 โ ๐7);
๐ฃ โ ๐ฃ ร ๐II(๐ก2,4 + ๐ก2,3 โ ๐1 โ ๐3, ๐7) ร ๐II(๐ก1,4 + ๐ก2,3 + ๐1 โ ๐3, ๐ก1,3 + ๐ก2,4 โ ๐1 + ๐3);
๐โฆ๐โง,๐ โ
{
(โ1)๐2
๐๐โ2๐โ1๐๐(๐๐โ๐โ1)!
(๐๐โ2๐)!๐!๐๐(๐๐ > 0)
(โ1)๐2|๐๐|โ2๐โ1(|๐๐|โ๐โ1)!
(|๐๐|โ2๐โ1)!๐!๐๐๐ ๐ ๐๐(๐๐ < 0)
1 ๐๐๐ ๐
๐๐๐ (๐ = 0,1โฏ , [|๐๐|โ๐๐
2]) ;
// The expansion coefficient of ๐๐๐ (๐๐๐) or ๐ ๐๐(|๐๐|๐) is denoted by ๐โฆ๐โง,๐;
๐ฃ โ ๐ฃ รโ ๐ดโฆ๐โง๐โฆ๐โง,๐๐,2๐โฆ๐โง,๐๐,1๐โฆ๐โง,๐๐,34๐=1 ;
๐ โ๐ + ๐ฃ ร ๐III(๐ก1,7, ๐ก1,5, ๐ก2,5, ๐ก1,6, ๐ก2,6, ๐ก2,4 + ๐ก2,3 โ ๐1 โ ๐3 + ๐7, ๐ก2,1 + ๐1 + ๐3 + 2๐5 + 2๐6 โ ๐7);
}}}}}}}}}}}}}}}}}}} ๐ ๐๐ก๐ข๐๐ (๐ โ ๐ ร ๐ด1๐ด2๐ด3๐ด4 ร โ1); } //End.
Appendix III: Radial Generalized Integral (Algorithm)
Algorithm name: radial integral ๐III(โ);
19
Input: (๏ฟฝโ๏ฟฝ );
Output: ๐ = โซ ๐๐1 โซ ๐๐2 โซ ๐1,2๐ 1๐1
๐ 2๐2๐ 3๐โ(๐ 4๐1+๐ 5๐2)sin๐ 6(๐ฝ1,2)cos
๐ 7(๐ฝ1,2)๐๐1,2๐1+๐2|๐1โ๐2|
+โ
0
+โ
0;
Algorithmic process:
๐III(๏ฟฝโ๏ฟฝ ){ ๐ผ๐ (0 โก ๐ 6 ๐๐๐ 2){
๐ โ โ โ โ(๐ 62)!(๐ 7+2๐1)!๐V(๐ 4,๐ 5,๐ 2+2๐2โ2๐3โ๐ 7โ2๐1,๐ 3+2๐3โ๐ 7โ2๐1,๐ 1+2๐ 7+4๐1โ2๐2)
(โ1)๐ 7+๐1โ๐2ร2๐ 7+2๐1(๐ 62โ๐1)!๐1!(๐ 7+2๐1โ๐2)!(๐2โ๐3)!๐3!
๐2๐3=0
๐ 7+2๐1๐2=0
๐ 62
๐1=0; }
๐ธ๐๐ ๐{
๐ โ โ โ โ โ(2๐2)!(
๐ 6โ1
2)!(๐ 7+2๐1+2๐2)!๐V(
๐ 4,๐ 5,๐ 2+2๐3โ2๐4โ๐ 7โ2๐1โ2๐2, ๐ 3+2๐4โ๐ 7โ2๐1โ2๐2,๐ 1+2๐ 7+4๐1+4๐2โ2๐3
)
(โ1)๐ 7+๐1โ๐3ร2๐ 7+2๐1+4๐2(1โ2๐2)(๐2!)2(๐ 6โ1
2โ๐1)!๐1!(๐ 7+2๐1+2๐2โ๐3)!(๐3โ๐4)!๐4!
๐3๐4=0
๐ 7+2๐1+2๐2๐3=0
+โ๐2=0
๐ 6โ1
2
๐1=0; }
๐ ๐๐ก๐ข๐๐ ๐; } //End.
Wherein, the definite integral of the intermediate function is denoted by
{
๐I(๐, ๐) = โซ sin๐(๐ฅ)cos๐(๐ฅ)
๐
0 ๐๐ฅ
๐II(๐, ๐) = โซ sin๐(๐ฅ)cos๐(๐ฅ)๐๐ฅ2๐
0
๐V(๏ฟฝโ๏ฟฝ ) = โซ ๐1๐ 3๐โ๐ 1๐1๐๐1 โซ ๐2
๐ 4๐โ๐ 2๐2๐๐2 โซ ๐1,2๐ 5๐๐1,2
๐1+๐2|๐1โ๐2|
+โ
0
+โ
0
.
References:
[1] Schrรถdinger E 1926 Ann. Phys.79 361.
[2] Schrรถdinger E 1926 Ann. Phys.79 489.
[3] Schrรถdinger E 1926 Ann. Phys.80 437.
[4] Schrรถdinger 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-Frรฉcon M, Grotendorst J 2006 Chem.Phys. 324 323.
[28] Hylleraas E A 1928 Z. Phys. 48 469.