theoretical calculation of atomic hyperfine structure zhao
TRANSCRIPT
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A new method for theoretical calculation of atomic hyperfine structure
Zhao Yukuo1)â 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.Aâ
E-mail: [email protected]
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 Ï 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
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is as follows (in atomic unit):
ð¢Ä§ð
ðð¡Ï = (ð¶ð¶ â ð + ï¿œÌï¿œð¶2 â
ð
ð)Ï (1).
Accordingly, (Dirac energy of hydrogen-like system) can be obtained:
ðžDH(ðð , ðð) =
1
ðŒ2(
ððâððâ1+â(ðð+1)2â(ðŒZ)2
â(ððâððâ1)2+(ðð+1)
2+2(ððâððâ1)â(ðð+1)2â(ðŒZ)2
â 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 Lambâs experimental value[12]. Then, (QED energy of hydrogen-like system) can be
obtained according to his calculation method of gradual development:
ðžQEDH (ðð , ðð , ðð , ðœð) = ðžD
H(ðð , ðð) + ðžLH(ðð , ðð , ðœð) + ðžM
H(ðð , ðð , ðð , ðœð) (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 ð¡, the light velocity is denoted
by ð¶, Dirac4 à 4 matrix is denoted by ð¶ and ï¿œÌï¿œ, the momentum operator is denoted by ð, the number of nuclear
charges is denoted by ð, the wave function of the single particle is denoted by Ï, the principal quantum number is
denoted by ðð = 1,2â¯, the azimuthal quantum number is denoted by ðð = 0,1⯠, (ðð â 1), the magnetic quantum
number is denoted by ðð = 0,±1⯠,±ðð, the spin quantum number is denoted by ðœð = 0 ðð 1, the fine structure
constant is denoted by ðŒ â1
137.036, and the intermediate function is denoted by
{
ðžL
H(ð, ð, ðœ) â {4(1â(â1)ðœ)â2ð
H Z4
ð3ðð(ð > 1 ððð ð = 0)
0 ððð ð
ðžMH(ð, ð,ð, ðœ) â {
3â1ðH ((2ðâ(â1)ðœ+1)(2ðâ(â1)ðœ+3)â(2ð+1)(2ð+3)â3)Z3
8(2ð+3)(2ð+1)2ð3ðð(ð = 0 ðð ð â odd number)
3â1ðH ((2ðâ(â1)ðœâ1)(2ðâ(â1)ðœ+1)â(2ðâ1)(2ð+1)â3)Z3
8(2ðâ1)(2ð+1)2ð3ððð ð
(4).
The ground state Lamb shift of hydrogen atoms is denoted by â1ðH = 0.5556ðŒ3, (22S1/2 â 22P1/2) state Lamb
shift is denoted by â2ðH = 0.4138ðŒ3, and the coordinate vector of electron ðð is denoted by ï¿œâï¿œ ð, as shown in Fig. 1.
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However, recently, we have discovered a new eigenequation, whose eigenvalue approximates the calculation
result of QED, i.e. (atom):
â (ð»ð â¡ â1
2âð2 â
ð
ððâð¿3(ð)
ðð2 â
ð¿2(ð)
ðð2ððð 2(ðð)
+ð¿1(ð)
ðð2ð ðð2(ðð)
+ â0.5
ðð,ð
ðð=1 ððð ðâ ð )ð
ð=1 ð¹ = ðžð¹ (5).
Wherein, the number of extranuclear electrons is denoted by ð, the eigenfunction (wave function) is denoted
by ð¹, the eigenvalue (system energy) is denoted by ðž, the Laplace operator is denoted by
âð2=
ð2
ðð¥ð2 +
ð2
ððŠð2 +
ð2
ðð§ð2 =
1
ðð2
ð
ððð(ðð
2 ð
ððð) +
1
ðð2 sin(ðð)
ð
ððð(sin(ðð)
ð
ððð) +
1
ðð2 sin2(ðð)
ð2
ððð2,
and a function related to ð is denoted by ð¿ð(ð), 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 ð¿ð(ð) 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 ð¹ð(ð) function
Suppose that the mass of the stator is denoted by ðð , the mass of the rotor is denoted by ðð and the
gravitation constant is denoted by ðº , (the gravitational potential) can be obtained according to Schwarzschild
metric[19]:
Fig. 1: Coordinate vector of electron ðð
ï¿œâï¿œ ð
ð¥
z
ðŠ
O
ðð
{
ðð = âð¥ð
2 + ðŠð2 + ð§ð
2
ðð = ðŽððððð (ð§ððð)
ðð = ðŽððððð (ð¥ð
ððsin(ðð))
ðð,ð = â(ð¥ð â ð¥ð)2+ (ðŠð â ðŠð)
2+ (ð§ð â ð§ð)
2
ðœð,ð = ðŽððððð áðð2 + ðð
2 â ðð,ð2
2ððððá
ðð,ð = ðŽððððð ácos(ðð) â cos(ðð)cos(ðœð,ð)
sin(ðð)sin(ðœð,ð)á
ï¿œâï¿œ ð
ðð,ð ðð
ðœð,ð
ðð,ð
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ððº(ð) â âðºðð ðð
ðâðº2ðð
2ðð
ð¶2ð2 (6).
âŽSuppose that the carried charge of the stator is denoted by ðð , the carried charge of the rotor is denoted by
ðð , the wlectrostatic force constant is denoted by ðŸ and a function of ðð is denoted by ð¿(ðð ) , (the
electromagnetic potential) can be obtained by analogy of Equation (6):
ðð¶(ð) â âðŸðð ðð
ðâð¿(ðð )ðð
ð2 (7).
So the motion of the charged system around the center generates a ððâ2 related magnetic potential (in the
spherical coordinate system):
ðð¿ð¹(ï¿œâï¿œ ð) = âð¿3(ð)
ðð2 â
ð¿2(ð)
ðð2ððð 2(ðð)
+ð¿1(ð)
ðð2ð ðð2(ðð)
(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.
âŽSuppose 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:
{
(
ð2
ðð2+ðð
2)ð·âŠðâ§(ð) = 0
áð2
ðð2+cos(ð)
sin(ð)
ð
ððâðð2+2ð¿1(ð)
sin2(ð)+
2ð¿2(ð)
cos2(ð)+ ð¿âŠðâ§(ð¿âŠð⧠+ 1)áð©âŠðâ§(ð) = 0
(ð2
ðð2+2
ð
ð
ðð+2ð
ðâð¿âŠðâ§(ð¿âŠðâ§+1)â2ð¿3(ð)
ð2+ 2ðžâŠðâ§
ð» )ð âŠðâ§(ð) = 0
(9).
Obtain:
{
ðžâŠðâ§
ð» = â1
2ðâŠðâ§2 ððð ðâŠð⧠=
ð
ððâððâ1
2+â(ð¿âŠðâ§+
1
2)2â2ð¿3(ð)
ΊâŠðâ§(ð) = {cos(ððð) ðð(ðð ⥠0)
sin(|ðð|ð) ððð ð
ð©âŠðâ§(ð) = ð ðð(â1)ðð+1âðð
2+2ð¿1(ð)(ð)â ðâŠðâ§,ðððð ðâŠðâ§â2ð(ð)
[ððâ|ðð|
2]
ð=0 (unnormalized)
ð âŠðâ§(ð) = â ðâŠðâ§,ðððâ
1
2+â(ð¿âŠðâ§+
1
2)2â2ð¿3(ð)ðâðâŠðâ§ð
ððâððâ1ð=0
(10).
Wherein, the atomic orbital is denoted by âŠð⧠= (ðð , ðð , ðð , ðœð , ðð; ðâŠðâ§), the parity quantum number is denoted
by ðð = {0 ðð 1 ðð(ðð = 0)
1 ððð ð , and the multinomial coefficient is denoted by
{
ðâŠðâ§,ð = {
1 ðð(ð = 0)
â(ðâŠðâ§â2ð+2)(ðâŠðâ§â2ð+1)+2ð¿2(ð)
2ð(2ð¿âŠðâ§+1â2ð)ðâŠðâ§,ðâ1 ððð ð
ðâŠðâ§,ð = {
1 ðð(ð = 0)
â2ðâŠðâ§(ððâððâð)
ðáð+â(2ð¿âŠðâ§+1)2â8ð¿3(ð)á
ðâŠðâ§,ðâ1 ððð ð
(11).
The intermediate function is denoted by
{
ðâŠð⧠= ðð â |ðð| +1
2â (â1)ðœðâ
1
4â 2ð¿2(ð)
ð¿âŠð⧠= ðâŠð⧠â (â1)ððâðð
2 + 2ð¿1(ð)
(12).
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âŽSuppose
{
ðžï¿œÌï¿œ,0,0,0,0
H = â1
ðŒ2+
1
ðŒ2â1â (
ðŒð
ï¿œÌï¿œ)2â3â1ð
H ð3
4ï¿œÌï¿œ3
ðžï¿œÌï¿œ,0,0,0,1H = â
1
ðŒ2+
1
ðŒ2â1â (
ðŒð
ï¿œÌï¿œ)2+â1ðH ð3
4ï¿œÌï¿œ3
ðžï¿œÌï¿œ,0,0,1,0H â ðžï¿œÌï¿œ+1,0,0,0,0
H =16â2ð
H ð4
(ï¿œÌï¿œ+1)4
ððð ï¿œÌï¿œ = [ðŒð + 1], can be obtained according to
ðžâŠðâ§ð» = â
1
2ðâŠðâ§2 in Equation (10)ïŒ
{
ð¿1(ð) =
(Î2(ð)âÎ3(ð)+16ââ(Î2(ð)âÎ3(ð)+16)2â64Î2(ð)+64Î1(ð))
2
2048
ð¿2(ð) =1
8â(16â2ð¿1(ð)âÎ2(ð)+Î1(ð))
2
1024ð¿1(ð)
ð¿3(ð) =1
8(2 â â1 â 8ð¿2(ð) â 2â2ð¿1(ð))
2âÎ1(ð)
8
(13).
Wherein, the intermediate function is denoted by
{
Î1(ð) =
á2ðŒðâ(2ï¿œÌï¿œâ1)â2+1.5â1ðH ðŒ2ð3ï¿œÌï¿œâ3â2â1â(ðŒð)2ï¿œÌï¿œâ2á
2
2+1.5â1ðH ðŒ2ð3ï¿œÌï¿œâ3â2â1â(ðŒð)2ï¿œÌï¿œâ2
Î2(ð) =á2ðŒðâ(2ï¿œÌï¿œâ1)â2â0.5â1ð
H ðŒ2ð3ï¿œÌï¿œâ3â2â1â(ðŒð)2ï¿œÌï¿œâ2á
2
2â0.5â1ðH ðŒ2ð3ï¿œÌï¿œâ3â2â1â(ðŒð)2ï¿œÌï¿œâ2
Î3(ð) = ((ï¿œÌï¿œ+1)2(2ï¿œÌï¿œ+1+âÎ1(ð))
â(ï¿œÌï¿œ+1)4â8â2ðH ð2(2ï¿œÌï¿œ+1+âÎ1(ð))
2â 2ï¿œÌï¿œ + 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:
ð¹âŠï¿œââï¿œ ⧠= â â (ðâŠðâ§(ï¿œâï¿œ ð)ðâŠðâ§(ï¿œâï¿œ ð) â (â1)ðâŠðâ§,âŠðâ§ðâŠðâ§(ï¿œâï¿œ ð)ðâŠðâ§(ï¿œâï¿œ ð))â ðâŠðâ§(ï¿œâï¿œ ð)
ððâ ð,ð
ðð=ð+1
ðâ1ð=1
ð . ð¡. â {ðâŠðâ§,âŠð⧠= 0
|(ðð+ðœðâ1)(ðð+ðœð)â(ðð+ðœðâ1)(ðð+ðœð)
2+ ðð + ðœð â ðð â ðœð| < 3
â¹ âšðâŠðâ§(ï¿œâï¿œ )|ðâŠðâ§(ï¿œâï¿œ )â© â 0 (15).
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Wherein, the hydrogen-like wave function is denoted by ðâŠðâ§(ï¿œâï¿œ ) = ðŽâŠðâ§ð·âŠðâ§(ð)ð©âŠðâ§(ð)ð âŠðâ§(ð), the electron
configuration is denoted by âŠï¿œââï¿œ ⧠= (âŠ1â§, âŠ2â§â¯ ), the normalization coefficient is denoted by
ðŽâŠð⧠=1
ââ«ð·âŠðâ§2 (ð)ð©âŠðâ§
2 (ð)ð âŠðâ§2 (ð) ðï¿œâï¿œ
, and the symmetry coefficient sis denoted by ðâŠðâ§,âŠð⧠= 0 ðð 1.
3.2 Energy functional
Suppose that the experimental value of the system is denoted by ðžâŠï¿œââï¿œ â§
ðð¥ð, the error rate is denoted by
íâŠï¿œââï¿œ ⧠=ðžâŠï¿œââï¿œ â§âðžâŠï¿œââï¿œ â§
ðð¥ð
|ðžâŠï¿œââï¿œ â§
ðð¥ð|à 100, and the energy functional minimization model for the helium-like systems is denoted by
ðžâŠï¿œââï¿œ â§He = ððð
1
ðŽâ âšð¹
âŠï¿œââï¿œ â§He|ð»ð|ð¹âŠï¿œââï¿œ â§
Heâ©ðð=1 (ð = 2 ððð ðŽ = âšð¹
âŠï¿œââï¿œ â§He |ð¹
âŠï¿œââï¿œ â§Heâ©) (16).
Then the maximum error rate of the helium-like system is ððð¥ {|íâŠï¿œââï¿œ â§|} â 0.96 (the specific calculation
process is similar to Hartree Fock method, omitted here).
Wherein, the Hamiltonian operator ð»ð is shown in Equation (5), the structure of the trial function ð¹âŠï¿œââï¿œ ⧠is
shown in Equation (15), and the correlation calculation results are shown in Table 1.
Table 1: Energy of helium atom (ðžâŠï¿œââï¿œ â§S = ððð
1
ðŽâ âšð¹
âŠï¿œââï¿œ â§He|â
1
2âð2 â
2
ðð+
0.5
ð1,2|ð¹
âŠï¿œââï¿œ â§Heâ©2
ð=1 ð. ð¢.)
ID ð1, ð1, ð1, ðœ1 ; ð2, ð2, ð2, ðœ2; ðâŠ1â§,âŠ2⧠ðžâŠï¿œââï¿œ â§He ðž
âŠï¿œââï¿œ â§S ðžðŒð·
Drake[18,26] íâŠï¿œââï¿œ â§
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
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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):
ððHF(ï¿œâï¿œ ð , ï¿œâï¿œ ð) =
ð1
ðð,ð+ð2ðð,ð
ðððð+ð3(ðð
2+ðð2)
ðððððð,ð+ð4ðð,ð
2
ðððð2 +
ð5(ðð+ðð)
ðððððð,ð+ð6ðððð
ðð,ð+ð7(ððð (ðð)ððð (ðð)+ð ðð(ðð)ð ðð(ðð)ððð (ððâðð))
ðððð (17).
Therefore, the Hamiltonian operator in the single particle approximation is (the spherical coordinate system):
ð»ðHF = â
1
2âð2 â
ð
ððâð¿3(ð)
ðð2 â
ð¿2(ð)
ðð2ððð 2(ðð)
+ð¿1(ð)
ðð2ð ðð2(ðð)
+ â ððHF(ï¿œâï¿œ ð, ï¿œâï¿œ ð)
ðð=1 ððð ðâ ð (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):
ðžâŠï¿œââï¿œ ⧠= ððð ðžðŒð·HF(âŠï¿œââï¿œ â§) =
1
ðŽâ âšð¹âŠï¿œââï¿œ â§|ð»ð
HF|ð¹âŠï¿œââï¿œ â§â©ðð=1 (19).
Moreover, according to the fitting technology of neural network and ððð ð¹(ð1, ð2,⯠ð7) = â |ðžðâðžð
NIST
ðžðNIST |ð=1 ,
it can be obtained(Fitting process, omitted):
ð1 = 0.47883387; ð2 = â0.01397390; ð3 = 0.00769582; ð4 = 0.00000713 ;
ð5 = 0.00231748; ð6 = 0.01837402; ð7 = â0.1701; (20).
4. Conclusion
4.1 Hyperfine structures of hydrogen atoms and ððð+ ions (ð †ð)
In order to verify the reasonableness of introducing the magnetic interaction potential ðð¿ð¹(ï¿œâï¿œ ð), 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 ðð , ðð , |ðð|, ðœð , ðð Îðžð Îðžðððžð· íð
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
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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 ðð , ðð , |ðð|, ðœð , ðð Îðžð Îðžðððžð·
íð
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, Îðžð = ðžâŠï¿œââï¿œ ⧠â ðžðððð¢ðð ð ð¡ðð¡ð , error rate íð =100(Îðžð
ððžð·âÎðžð)
Îðžðððžð· ðð
100(ÎðžðNISTâÎðžð)
ÎðžðNIST , the calculation
method of ðžâŠï¿œââï¿œ ⧠is shown in Equation (10) or (19), and the calculation method of ðžðððžð·
is shown in Equation (3).
Next, in Table 2 and Table 3, Lamb shift= {Îðž2 â Îðž1 = 0.0000002159 ðð 0.1681208972Îðž5 â Îðž3 = 0.0000001608 ðð 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 3â€IDâ€9. For example, in the hydrogen atoms,
Îðž4âÎðž3
Îðž4ððžð·
âÎðž3ððžð· â 0.14 ððð
Îðž4âÎðž3
ðžMH(2,0,0,1)âðžM
H(2,0,0,0)â 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]
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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
Betheâs 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 (ð1, ð1, ð1, ðœ1, ð1) = (1,0,0,0,0).
ID ð2, ð2, |ð2|, ðœ2, ð2; ðâŠ1â§,âŠ2⧠ð1 ð2 Îðžð ÎðžðNIST[18]
íð
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
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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 {
(ð1, ð1, ð1, ðœ1, ð1) = (1,0,0,0,0)
(ð2, ð2, ð2, ðœ2, ð2) = (1,0,0,0,1)ðâŠ1â§,âŠ2⧠= 1
.
ID ð3, ð3, |ð3|, ðœ3, ð3; ðâŠ1â§,âŠ3â§, ðâŠ2â§,âŠ3⧠ð1 ð2 ð3 Îðžð ÎðžðNIST[18]
íð
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
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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
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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 ð1 ð2 ð3 ðžðððð¢ðð ð ð¡ðð¡ð ðžðððð¢ðð ð ð¡ðð¡ðNIST[18]
íð
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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 ð2 in the helium-like structure is fixed, the following
equation can be obtained:
(â1
2â12 â
ð
ð1+
1
ð1,2)ð¹ = (ðž +
ð
ð2)ð¹ (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 âð â ð â¹ ðð â ðð 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 |íð|% is
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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
ðâŠðâ§,âŠðâ§. Therefore, we will report the theoretical calculation in this aspect in the subsequent research articles.
Appendix I: Energy Functional (Algorithm)
Algorithm name: energy functional ðžðŒð·HF(â);
Input: âŠï¿œââï¿œ â§; // Configuration âŠï¿œââï¿œ ⧠= (âŠ1â§, âŠ2â§â¯ ) ððð âŠð⧠= (ðð , ðð , ðð , ðœð , ðð; ðâŠðâ§).
Output: ð =1
ðŽÃ â âšð¹âŠï¿œââï¿œ â§|ð»ð
HF|ð¹âŠï¿œââï¿œ â§â©ðð=1 ; // ðŽ = âšð¹âŠï¿œââï¿œ â§|ð¹âŠï¿œââï¿œ â§â©.
Algorithmic process:
ðžðŒð·HF(âŠï¿œââï¿œ â§) {
Initially assigned values:
ðŽ â 0; ð â 0; ð â 0;
â1,1 â ð1; â2,1 â ð2; â3,1 â ð3;â4,1 â ð3; â5,1 â ð4; â6,1 â ð5;â7,1 â ð5; â8,1 â ð6;
â1,2 â 0; â2,2 â â1; â3,2 â â1;â4,2 â 1; â5,2 â â1;â6,2 â â1;â7,2 â 0; â8,2 â 1;
â1,3 â 0; â2,3 â â1; â3,3 â 1;â4,3 â â1; â5,3 â â2;â6,3 â 0;â7,3 â â1; â8,3 â 1;
â1,4 â â1;â2,4 â 1;â3,4 â â1;â4,4 â â1; â5,4 â 2; â6,4 â â1;â7,4 â â1; â8,4 â â1;
ð¹ðð(ð â 1; ð †8; ð â ð + 1){âð,12 â âð,11 â âð,10 â âð,9 â âð,8 â âð,7 â âð,6 â âð,5 â 0; }
â9,1 â ð7; â9,3 â â9,2 â â1; â9,12 â â9,11 â â9,10 â â9,9 â â9,8 â â9,7 â â9,4 â 0; â9,6 â â9,5 â 1;
â10,1 â ð7; â10,3 â â10,2 â â1; â10,12 â â10,11 â â10,6 â â10,5 â â10,4 â 0;
â11,1 â ð7; â11,3 â â11,2 â â1; â11,10 â â11,9 â â11,6 â â11,5 â â11,4 â 0;
â10,10 â â10,9 â â10,8 â â10,7 â â11,12 â â11,11 â â11,8 â â11,7 â 1;
ð¹ðð(ð â 1; ð †ð; ð â ð + 1){ ð¹ðð(ð â 1; ð †ð; ð â ð + 1){
ðâŠðâ§,âŠð⧠â {0 ðð 1 ðð(âšðâŠðâ§(ï¿œâï¿œ )|ðâŠðâ§(ï¿œâï¿œ )â© â 0)
1 ððð ð ïŒ
ð¢ð,ð â âšðâŠðâ§|ðâŠðâ§â©; ðð,ð â âšðâŠðâ§|1
ð|ðâŠðâ§â© ; }} //ðâŠð⧠= ðâŠðâ§(ï¿œâï¿œ );
Correlation calculation of potential energy and repulsive energy:
ð¹ðð(ð1 â 1; ð1 †ð â 1; ð1 â ð1 + 1){ ð¹ðð(ð1 â ð1 + 1; ð1 †ð; ð1 â ð1 + 1){
ð¹ðð(ð2 â 1; ð2 †ð â 1; ð2 â ð2 + 1){ ð¹ðð(ð2 â ð2 + 1; ð2 †ð; ð2 â ð2 + 1){
ð£1 â â(â1)ðâŠð2â§,âŠð2â§;ð£2 â â(â1)ðâŠð1â§,âŠð1â§; ð£3 â (â1)ðâŠð1â§,âŠð1â§+ðâŠð2â§,âŠð2â§;
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Normalization coefficient:
ðŽ â ðŽ + 1 + ð£1ð¢ð2,ð22 + ð£2ð¢ð1,ð1
2 ;
ðŒð(ð2 = ð1 ððð ð2 = ð1){ðŽ â ðŽ + ð£3; } ðžðð ð ðð(ð2 = ð1 ððð ð2 â ð1){ðŽ â ðŽ + ð£3ð¢ð1,ð2ð¢ð1,ð1ð¢ð1,ð2; }
ðžðð ð ðð(ð2 = ð1){ðŽ â ðŽ + ð£3ð¢ð1,ð1ð¢ð1,ð2ð¢ð1,ð2; } ðžðð ð ðð(ð2 = ð1){ðŽ â ðŽ + ð£3ð¢ð1,ð2ð¢ð1,ð1ð¢ð2,ð1; }
ðžðð ð ðð(ð2 = ð1){ðŽ â ðŽ + ð£3ð¢ð1,ð1ð¢ð1,ð2ð¢ð2,ð1; } ðžðð ð{ðŽ â ðŽ + ð£3ð¢ð1,ð12 ð¢ð2,ð2
2 ; }
Potential energy:
ð¹ðð(ð3 â 1; ð3 †ð; ð3 â ð3 + 1){
ð£4 â ðâŠð3⧠áðð3 â ðð3 â1
2+â(ð¿âŠð3⧠+
1
2)2â 2ð¿3(ð)á â ð ; ð â ð + ð£4ðð3,ð3;
ðŒð(ð3 = ð2 ðð ð2){ ð â ð + ð£1ð£4ð¢ð2,ð2ðð2,ð2; } ðžðð ð{ð â ð + ð£1ð£4ð¢ð2,ð22 ðð3,ð3; }
ðŒð(ð3 = ð1 ðð ð1){ ð â ð + ð£2ð£4ð¢ð1,ð1ðð1,ð1; } ðžðð ð{ð â ð + ð£2ð£4ð¢ð1,ð12 ðð3,ð3; }
ðŒð(ð2 = ð1){ðŒð(ð2 = ð1){ð â ð + ð£3ð£4ðð3,ð3; } ðžðð ð{ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð2; }
ðžðð ð ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð2ð¢ð1,ð2ðð3,ð1; } ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð1; }
ðžðð ð{ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ð¢ð1,ð2ðð3,ð3; }}}
ðžðð ð ðð(ð2 = ð1){ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð2ð¢ð1,ð2ðð3,ð1; }
ðžðð ð ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð2; } ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð1; }
ðžðð ð{ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ð¢ð1,ð2ðð3,ð3; }}
ðžðð ð{ðð(ð2 = ð1){ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð2; }
ðžðð ð ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð2,ð1ð¢ð1,ð2ðð3,ð1; } ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð1; }
ðžðð ð{ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ð¢ð1,ð2ðð3,ð3; }}
ðžðð ð ðð(ð2 = ð1){ ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð2,ð1ð¢ð1,ð2ðð3,ð1; }
ðžðð ð ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð2; } ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ðð3,ð1; }
ðžðð ð{ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð1,ð2ð¢ð1,ð2ðð3,ð3; }}
ðžðð ð{ ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð2,ð22 ðð3,ð1; }
ðžðð ð ðð(ð3 = ð1){ð â ð + ð£3ð£4ð¢ð1,ð1ð¢ð2,ð22 ðð3,ð1; } ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð1
2 ð¢ð2,ð2ðð3,ð2; }
ðžðð ð ðð(ð3 = ð2){ð â ð + ð£3ð£4ð¢ð1,ð12 ð¢ð2,ð2ðð3,ð2; }ðžðð ð{ð â ð + ð£3ð£4ð¢ð1,ð1
2 ð¢ð2,ð22 ðð3,ð3; }}}}
Repulsive energy between electrons (monocentric double-electron coupling integral is shown in Appendix II below):
ð¹ðð(ð3 â 1; ð3 < ð; ð3 â ð3 + 1){ð¹ðð(ð3 â ð3 + 1; ð3 †ð; ð3 â ð3 + 1){ð¹ðð(ð â 1; ð †11; ð â ð + 1){
ð â ð + ðŒII(âŠð3â§, âŠð3â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); //ï¿œââï¿œ ð = (âð,1, âð,2â¯âð,12).
ðŒð(ð2 = ð1 ððð ð2 = ð1){ð â ð + ð£3ðŒII(âŠð3â§, âŠð3â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð2 â ð1 ððð ð2 â ð1 ððð ð2 â ð1 ððð ð2 â ð1) {
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ðŒð(ð3 = ð1 ððð ð3 = ð1){ð â ð + ð£3ð¢ð2,ð22 ðŒII(âŠð1â§, âŠð1â§, âŠð1â§, âŠð1â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð2 ððð ð3 = ð2){ð â ð + ð£3ð¢ð1,ð12 ðŒII(âŠð2â§, âŠð2â§, âŠð2â§, âŠð2â§; ï¿œââï¿œ ð); }
ðžðð ð ðð((ð3 = ð1 ðð ð3 = ð1) ððð (ð3 = ð2 ðð ð3 = ð2)){ð â ð + ð£3ð¢ð1,ð1ð¢ð2,ð2ðŒII(âŠð1â§, âŠð1â§, âŠð2â§, âŠð2â§; ï¿œââï¿œ ð); }
ðžðð ð ðð((ð3 = ð2 ðð ð3 = ð2) ððð (ð3 = ð1 ðð ð3 = ð1)){ð â ð + ð£3ð¢ð1,ð1ð¢ð2,ð2ðŒII(âŠð1â§, âŠð1â§, âŠð2â§, âŠð2â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð1 ðð ð3 = ð1){ð â ð + ð£3ð¢ð1,ð1ð¢ð2,ð22 ðŒII(âŠð1â§, âŠð1â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð2 ðð ð3 = ð2){ð â ð + ð£3ð¢ð2,ð2ð¢ð1,ð12 ðŒII(âŠð2â§, âŠð2â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð1 ðð ð3 = ð1){ð â ð + ð£3ð¢ð1,ð1ð¢ð2,ð22 ðŒII(âŠð1â§, âŠð1â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð2 ðð ð3 = ð2){ð â ð + ð£3ð¢ð2,ð2ð¢ð1,ð12 ðŒII(âŠð2â§, âŠð2â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð{ð â ð + ð£3ð¢ð1,ð12 ð¢ð2,ð2
2 ðŒII(âŠð3â§, âŠð3â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
} ð¹ðð(ð â 1; ð < 3; ð â ð + 1){
ðŒð(ð = 1){ð¡1 â ð2; ð¡2 â ð2; } ðžðð ð {ð¡1 â ð1; ð¡2 â ð1; }
ðŒð(ð3 = ð¡1 ððð ð3 = ð¡2){ð â ð + ð£ð ðŒII(âŠð¡1â§, âŠð¡2â§, âŠð¡1â§, âŠð¡2â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡1 ðð (ð3 â ð¡1 ððð ð3 = ð¡2)){ð â ð + ð£ð ð¢ð¡1,ð¡2ðŒII(âŠð3â§, âŠð3â§, âŠð¡1â§, âŠð¡2â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡2 ðð (ð3 = ð¡1 ððð ð3 â ð¡2)){ð â ð + ð£ð ð¢ð¡1,ð¡2ðŒII(âŠð¡1â§, âŠð¡2â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð{ð â ð + ð£ð ð¢ð¡1,ð¡22 ðŒII(âŠð3â§, âŠð3â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
} ð¹ðð(ð â 1; ð < 7; ð â ð + 1){
ðð(ð = 1 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð2; ð¡4 â ð1; ð¡5 â ð1; ð¡6 â ð2; ð¡7 â ð1; ð¡8 â ð2; ð¡9 â ð1; ð¡10 â ð1; ð¡11 â ð1; }
ðð(ð = 2 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð1; ð¡4 â ð1; ð¡5 â ð2; ð¡6 â ð1; ð¡7 â ð1; ð¡8 â ð1; ð¡9 â ð1; ð¡10 â ð1; ð¡11 â ð2; }
ðð(ð = 3 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð1; ð¡4 â ð1; ð¡5 â ð2; ð¡6 â ð1; ð¡7 â ð2; ð¡8 â ð1; ð¡9 â ð1; ð¡10 â ð1; ð¡11 â ð2; }
ðð(ð = 4 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð1; ð¡4 â ð2; ð¡5 â ð1; ð¡6 â ð1; ð¡7 â ð2; ð¡8 â ð1; ð¡9 â ð2; ð¡10 â ð1; ð¡11 â ð1; }
ðð(ð = 5 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð2; ð¡4 â ð1; ð¡5 â ð1; ð¡6 â ð2; ð¡7 â ð1; ð¡8 â ð1; ð¡9 â ð1; ð¡10 â ð1; ð¡11 â ð2; }
ðð(ð = 6 ){ð¡1 â ð2; ð¡2 â ð1; ð¡3 â ð1; ð¡4 â ð2; ð¡5 â ð1; ð¡6 â ð1; ð¡7 â ð1; ð¡8 â ð2; ð¡9 â ð1; ð¡10 â ð1; ð¡11 â ð2; }
ðð(ð¡1 = ð¡2 ððð ð¡3 < ð¡4 < ð¡5){
ðŒð(ð3 = ð¡3 ððð ð3 = ð¡4){ð â ð + ð£3ð¢ð¡10,ð¡11ðŒII(âŠð¡6â§, âŠð¡7â§, âŠð¡8â§, âŠð¡9â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡3 ððð ð3 = ð¡5){ð â ð + ð£3ð¢ð¡8,ð¡9ðŒII(âŠð¡6â§, âŠð¡7â§, âŠð¡10â§, âŠð¡11â§; ï¿œââï¿œ ð); }
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ðžðð ð ðð(ð3 = ð¡4 ððð ð3 = ð¡5){ð â ð + ð£3ð¢ð¡6,ð¡7ðŒII(âŠð¡8â§, âŠð¡9â§, âŠð¡10â§, âŠð¡11â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡3){ð â ð + ð£3ð¢ð¡8,ð¡9ð¢ð¡10,ð¡11ðŒII(âŠð¡6â§, âŠð¡7â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡4){ð â ð + ð£3ð¢ð¡6,ð¡7ð¢ð¡10,ð¡11ðŒII(âŠð¡8â§, âŠð¡9â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡5){ð â ð + ð£3ð¢ð¡6,ð¡7ð¢ð¡8,ð¡9ðŒII(âŠð¡10â§, âŠð¡11â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡3){ð â ð + ð£3ð¢ð¡8,ð¡9ð¢ð¡10,ð¡11ðŒII(âŠð¡6â§, âŠð¡7â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡4){ð â ð + ð£3ð¢ð¡6,ð¡7ð¢ð¡10,ð¡11ðŒII(âŠð¡8â§, âŠð¡9â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð ðð(ð3 = ð¡5){ð â ð + ð£3ð¢ð¡6,ð¡7ð¢ð¡8,ð¡9ðŒII(âŠð¡10â§, âŠð¡11â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
ðžðð ð{ð â ð + ð£3ð¢ð¡6,ð¡7ð¢ð¡8,ð¡9ð¢ð¡10,ð¡11ðŒII(âŠð3â§, âŠð3â§, âŠð3â§, âŠð3â§; ï¿œââï¿œ ð); }
}}}}}}}}}
Total energy:
ð âð+ 2ð
ðŽ; ð¹ðð(ð â 1; ð †ð; ð â ð + 1) {ð â ð â
1
2ðð2; }
ð ðð¡ð¢ðð ð; } //End.
Appendix II: Monocentric Double-electron Coupling Integral (Algorithm)
Lemma[26]: six-dimensional integral element ðï¿œâï¿œ 1ðï¿œâï¿œ 2 = ð1ð2ð1,2ð ðð(ð1) ðð1ðð2ðð1,2ðð1ðð1ðð1,2.
Proposition (proof, omitted):
{
cos(ð2) = cos(ð1)cos(ðœ1,2) + sin(ð1)cos(ð1,2)sin(ðœ1,2)
cos(ð2) =cos(ðœ1,2)cos(ð1)
sin(ð2)sin(ð1)âcos(ð2)cos(ð1)cos(ð1)
sin(ð2)sin(ð1)+sin(ðœ1,2)sin(ð1,2)sin(ð1)
sin(ð2)
sin(ð2) =cos(ðœ1,2)sin(ð1)
sin(ð2)sin(ð1)âcos(ð2)cos(ð1)sin(ð1)
sin(ð2)sin(ð1)+sin(ðœ1,2)sin(ð1,2)cos(ð1)
sin(ð2)
.
Therefore, the monocentric double-electron coupling integral algorithm we adopted is shown below (the
representation of coordinate vector ï¿œâï¿œ ð is shown in Figure 1):
Algorithm name: monocentric double-electron coupling integral ðŒII(â);
Input: (âŠ1â§, âŠ2â§, âŠ3â§, âŠ4â§; ï¿œââï¿œ );
Output: ð = âšðâŠ1â§(ï¿œâï¿œ 1)ðâŠ2â§(ï¿œâï¿œ 1)|â1ð1â2ð2
â3ð1,2â4ð(ï¿œâï¿œ 1, ï¿œâï¿œ 2; ï¿œââï¿œ )|ðâŠ3â§(ï¿œâï¿œ 2)ðâŠ4â§(ï¿œâï¿œ 2)â©;
// ð(ï¿œâï¿œ 1, ï¿œâï¿œ 2; ï¿œââï¿œ ) = ððð â5(ð1)ððð
â6(ð2)ð ððâ7(ð1)ð ðð
â8(ð2)ððð â9(ð1)ððð
â10(ð2)ð ððâ11(ð1)ð ðð
â12(ð2).
Algorithmic process(The algorithm is not optimized due to length):
ðŒII(âŠ1â§, âŠ2â§, âŠ3â§, âŠ4â§; ï¿œââï¿œ ){ ð â 0;
ð¹ðð(ð1,1 â 0; ð1,1 < ð1 â ð1; ð1,1 â ð1,1 + 1){ ð¹ðð (ð1,2 â 0; ð1,2 †[ð1â|ð1|
2] ; ð1,2 â ð1,2 + 1) {
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18
ð1 â {0 ðð(ð1 ⥠0)
1 ððð ð ; ð¹ðð (ð1,3 â 0; ð1,3 †[
|ð1|âð1
2] ; ð1,3 â ð1,3 + 1) {
â® â®
ð¹ðð(ð4,1 â 0; ð4,1 < ð4 â ð4; ð4,1 â ð4,1 + 1){ ð¹ðð (ð4,2 â 0; ð4,2 †[ð4â|ð4|
2] ; ð4,2 â ð4,2 + 1) {
ð4 â {0 ðð(ð4 ⥠0)
1 ððð ð ; ð¹ðð (ð4,3 â 0; ð4,3 †[
|ð4|âð4
2] ; ð4,3 â ð4,3 + 1) {
ð¡1,1 â ð1 + ðœ1 â |ð1| â 2ð1,2 + ð2 + ðœ2 â |ð2| â 2ð2,2 + â5; ð¡1,2 â |ð1| + |ð2| + â7;
ð¡2,1 â ð3 + ðœ3 â |ð3| â 2ð3,2 + ð4 + ðœ4 â |ð4| â 2ð4,2 + â6; ð¡2,2 â |ð3| + |ð4| + â8;
ð¡1,3 â |ð1| â ð1 â 2ð1,3 + |ð2| â ð2 â 2ð2,3 + â9; ð¡1,4 â ð1 + ð2 + â11;
ð¡2,3 â |ð3| â ð3 â 2ð3,3 + |ð4| â ð4 â 2ð4,3 + â10; ð¡2,4 â ð3 + ð4 + â12;
ð¡1,5 â ð1 + ðœ1 + ð1,1 + ð2 + ðœ2 + ð2,1 + â2 + 1; ð¡1,6 â ð1 + ð2; ð¡1,7 â â4 + 1;
ð¡2,5 â ð3 + ðœ3 + ð3,1 + ð4 + ðœ4 + ð4,1 + â3 + 1; ð¡2,6 â ð3 + ð4;
ðŒð (ð¡1,2 < ð¡2,2) {ð¹ðð(ð â 1; ð †6; ð â ð + 1){ ð£ â ð¡1,ð; ð¡1,ð â ð¡2,ð; ð¡2,ð â ð£; }} ð¡1,2 â ð¡1,2 + 1;
ðŒð (0 â¡ (ð¡2,2 â ð¡2,4 â ð¡2,3) ððð 2){ð â 0; ð â 0; } ðžðð ð {ð â 2; ð â 1; }
ð¹ðð(ð1 â 0; ð1 †ð¡2,4; ð1 â ð1 + 1){ ð¹ðð(ð2 â 0; ð2 †ð1; ð2 â ð2 + 1){
ð¹ðð(ð3 â 0; ð3 †ð¡2,3; ð3 â ð3 + 1){ ð¹ðð(ð4 â 0; ð4 †ð3; ð4 â ð4 + 1){
ð¹ðð (ð5 â 0; ð5 â€ð¡2,2âð¡2,4âð¡2,3âð
2; ð5 â ð5 + 1) { ð¹ðð(ð6 â 0; ð6 †ð ; ð6 â ð6 + 1){
ð¹ðð(ð7 â 0; ð7 †ð¡2,1 + ð2 + ð4 + 2ð5 + 2ð6; ð7 â ð7 + 1){
ð£ â(â1)ð2+ð4+ð5ð¡2,3!ð¡2,4!(
ð¡2,2âð¡2,4âð¡2,3âð
2)!(2ð6)!(ð¡2,1+ð2+ð4+2ð5+2ð6)!
4ð6(1â2ð6)(ð6!)2ð2!(ð1âð2)!(ð¡2,4âð1)!ð4!(ð3âð4)!(ð¡2,3âð3)!ð5!(
ð¡2,2âð¡2,4âð¡2,3âð
2âð5)!ð7!(ð¡2,1+ð2+ð4+2ð5+2ð6âð7)!
;
ð£ â ð£ à ðI(ð¡1,2 + ð7 â ð1 â ð3, ð¡1,1 + ð¡2,1 + 2ð2 + 2ð4 + 2ð5 + 2ð6 â ð7);
ð£ â ð£ à ðII(ð¡2,4 + ð¡2,3 â ð1 â ð3, ð7) à ðII(ð¡1,4 + ð¡2,3 + ð1 â ð3, ð¡1,3 + ð¡2,4 â ð1 + ð3);
ðâŠðâ§,ð â
{
(â1)ð2
ððâ2ðâ1ðð(ððâðâ1)!
(ððâ2ð)!ð!ðð(ðð > 0)
(â1)ð2|ðð|â2ðâ1(|ðð|âðâ1)!
(|ðð|â2ðâ1)!ð!ððð ð ðð(ðð < 0)
1 ððð ð
ððð (ð = 0,1⯠, [|ðð|âðð
2]) ;
// The expansion coefficient of ððð (ððð) or ð ðð(|ðð|ð) is denoted by ðâŠðâ§,ð;
ð£ â ð£ Ãâ ðŽâŠðâ§ðâŠðâ§,ðð,2ðâŠðâ§,ðð,1ðâŠðâ§,ðð,34ð=1 ;
ð âð + ð£ à ðIII(ð¡1,7, ð¡1,5, ð¡2,5, ð¡1,6, ð¡2,6, ð¡2,4 + ð¡2,3 â ð1 â ð3 + ð7, ð¡2,1 + ð1 + ð3 + 2ð5 + 2ð6 â ð7);
}}}}}}}}}}}}}}}}}}} ð ðð¡ð¢ðð (ð â ð à ðŽ1ðŽ2ðŽ3ðŽ4 à â1); } //End.
Appendix III: Radial Generalized Integral (Algorithm)
Algorithm name: radial integral ðIII(â);
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Input: (ï¿œâï¿œ );
Output: ð = â« ðð1 â« ðð2 â« ð1,2ð 1ð1
ð 2ð2ð 3ðâ(ð 4ð1+ð 5ð2)sinð 6(ðœ1,2)cos
ð 7(ðœ1,2)ðð1,2ð1+ð2|ð1âð2|
+â
0
+â
0;
Algorithmic process:
ðIII(ï¿œâï¿œ ){ ðŒð (0 â¡ ð 6 ððð 2){
ð â â â â(ð 62)!(ð 7+2ð1)!ðV(ð 4,ð 5,ð 2+2ð2â2ð3âð 7â2ð1,ð 3+2ð3âð 7â2ð1,ð 1+2ð 7+4ð1â2ð2)
(â1)ð 7+ð1âð2Ã2ð 7+2ð1(ð 62âð1)!ð1!(ð 7+2ð1âð2)!(ð2âð3)!ð3!
ð2ð3=0
ð 7+2ð1ð2=0
ð 62
ð1=0; }
ðžðð ð{
ð â â â â â(2ð2)!(
ð 6â1
2)!(ð 7+2ð1+2ð2)!ðV(
ð 4,ð 5,ð 2+2ð3â2ð4âð 7â2ð1â2ð2, ð 3+2ð4âð 7â2ð1â2ð2,ð 1+2ð 7+4ð1+4ð2â2ð3
)
(â1)ð 7+ð1âð3Ã2ð 7+2ð1+4ð2(1â2ð2)(ð2!)2(ð 6â1
2âð1)!ð1!(ð 7+2ð1+2ð2âð3)!(ð3âð4)!ð4!
ð3ð4=0
ð 7+2ð1+2ð2ð3=0
+âð2=0
ð 6â1
2
ð1=0; }
ð ðð¡ð¢ðð ð; } //End.
Wherein, the definite integral of the intermediate function is denoted by
{
ðI(ð, ð) = â« sinð(ð¥)cosð(ð¥)
ð
0 ðð¥
ðII(ð, ð) = â« sinð(ð¥)cosð(ð¥)ðð¥2ð
0
ðV(ï¿œâï¿œ ) = â« ð1ð 3ðâð 1ð1ðð1 â« ð2
ð 4ðâð 2ð2ðð2 â« ð1,2ð 5ðð1,2
ð1+ð2|ð1âð2|
+â
0
+â
0
.
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