Unorthodox dimensional interpolations for He, Li, Be atoms and hydrogen molecule

We present a simple interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain the $D=3$ ground-state energies of atoms and molecules. For atoms, these limits are linked by first-order perturbation terms of electron-electron interactions. This unorthodox approach is illustrated by ground-states for two, three, and four electron atoms, with modest effort to obtain fairly accurate results. Also, we treat the ground-state of H$_2$ over a wide range of the internuclear distance R, and compares well with the standard exact results from the Full Configuration Interaction method. Similar dimensional interpolations may be useful for complex many-body systems.


Introduction
Dimensional scaling, as applied to chemical physics, offers promising computational strategies and heuristic perspectives to study electronic structures and obtain energies of atoms, molecules and extended systems. [1][2][3][4] Taking a spatial dimension other than D = 3 can make a problem much simpler and then use perturbation theory or other techniques to obtain an approximate result for D = 3. Years ago, a D-scaling technique used with quantum chromodynamics 5 was prompted for helium. [2][3][4] The approach began with the D → ∞ limit and added terms in powers of δ = 1/D. It was arduous and asymptotic but by summation techniques attained very high accuracy for D = 3. 6 Other dimensional scaling approaches were extended to N-electron atoms, 7 renormalization with 1/Z expansions, 8 random walks, 9 interpolation of hard sphere virial coefficients, 10 resonance states 11 and dynamics of manybody systems in external fields. 12,13 Recently, a simple analytical interpolation formula emerged using both the D = 1 and D → ∞ limits for helium. 14 It makes use of only the dimensional dependence of a hydrogen atom, together with the exactly known first-order perturbation terms with λ = 1/Z for the dimensional limits of the electron-electron 1/r 12 interaction. In the D = 1 limit, the Columbic potentials are replaced by delta functions in appropriately scaled coordinates. 15 In the D → ∞ limit, the electrons assume positions fixed relative to another and to the nucleus, with wave functions replaced by delta functions. 16 Then at D = 3, the ground state energy of helium 3 can be obtained by linking 1 and ∞ together with the first-order perturbation coefficients (1) 1 and (1) ∞ of the 1/Z expansion. The first-order terms actually provide much of the dimension dependence. This article exhibits the applicability of an unorthodox formula, a blend of dimensions with first-order perturbations, to more complex many-body systems.
We outline the following sections: in 2 the interpolation formula; in 3 treat helium; in 4 lithium; in 5 beryllium; in 6 hydrogen molecule. Each atom section 3-5 has four subsections: A for D = 1; B for D → ∞; C for (1) D , the first-order perturbation terms; D for 3 , the ground-state energy at D = 3 is obtained from the interpolation formula. For the hydrogen molecule section 6, the subsections deal how the internuclear distance R varies in the D = 1 and D → ∞ dimensions and mesh into D = 3. Finally, in 7 we comment on prospects for blending dimensional limits to serve other many-body problems.

Dimensional interpolation
For dimensional scaling of atoms and molecules the energy erupts to infinity as D → 1 and vanishes as D → ∞. Hence, we adopt scaled units (with hartree atomic units) whereby , so the reduced energy D remains finite in both limits. When expressed in a 1/Z perturbation expansion, the reduced energy is given by with λ = 1/Z, where Z is the total nuclear charge of the corresponding atom. The first-order perturbation coefficient is (1, 6): (1) It represents the expectation value, 1 r 12 , of the electron-electron repulsion evaluated with the zeroth-order hydrogenic wave function, exp(−r 1 − r 2 ). Accordingly, We aim to illustrate the interpolation formula more fully, presenting results with modest calculations having respectable accuracy for two, three, and four electrons.
For the hydrogen molecule, a different scaling scheme will be used and illustrated. The rescaling of distances is: An approximation for D = 3 (where R = R ) emerges: on interpolating linearly between the dimensional limits, developed by Loeser in Refs. [17][18][19] 3 Two-electrons: Helium The formula worked very well for D = 3, helium with λ = 1/2: The

One-dimension: D=1
We will calculate the ground-state energy of the Hamiltonian operator using the variational principle. It is less accurate than Ref., 15 but much easier to deal with two and more electrons. 20 The Hamiltonian with electrons in delta functions is: such that and obtain ξ 0 = 0.875, which put into Eq (15) gives the ground-state energy, 1 = −0.765625. This result is found in Refs., [20][21][22] but it is approximated by 2.9% since noted the exact value is 1 = −0.788843.

Infinite-dimension: D → ∞
At large-D limit, the effective ground state Hamiltonian for a two electron atom, with inter-electronic correlation can be written as: with J(r 1 , r 2 , θ) = 1 for an inter-electronic angle θ.
We minimize the above effective-Hamiltonian with respect to the parameters r 1 , r 2 , and θ respectively, and obtain the corresponding ground state energy to be: ∞ = −0.684442 (see Table 1 in, 14 and 22 ).

First-order perturbations:
(1) D In two-electron atom, with nuclear charge Z, the exact Hamiltonian in D-dimension using atomic units can be written as: where the Laplacian operator 2 r in D-dimension is defined as: For helium-like atoms we consider the two electrons are in a 1s-like state with spatial part being symmetric (both electrons are in the same state) and the spin part in the antisymmetric spin singlet. The spatial part of the electronic wave function can be written as: where the normalized wave functions χ 1 (r 1 ) and χ 2 (r 2 ) are defined as: and The normalization constant N is calculated as: with is the surface area of an unit sphere in D-dimension.
In D-dimension, with the above wave functions, we obtain the following first-order coefficient: 14 As shown in Eq.(2) and for D = 1, 3, ∞ respectively (1)
In conventional quantum chemistry textbooks treating D = 3 helium, the electronelectron interaction, 1/r 12 , is evaluated by first-order perturbation theory. The result is 3 = −0.687529 with accuracy of 5.29%.

Three-electrons: Lithium
The ground-state of the lithium atom had been calculated a long ago by using the variational method with complicated wave functions. [23][24][25] Here we present the interpolation formula, using the D = 1 and D = ∞ limits and the first-order perturbation terms. For the ground-state of the lithium atom our formula gave 3 = −0.839648, with approximation 1.04% compared the exact result 3 = −0.830896. 26

One-dimension: D=1
In a three-electron atom, with nuclear charge Z, the exact Hamiltonian in one-dimension using atomic units can be written as: with λ = 1/Z.
In lithium atom we consider two electrons are in 1s state and third electron is in a 2s state, with spatial part being symmetric (both electrons are in the same state) and the spin part in the antisymmetric state. We write spatial part of the electronic wave function as: The two normalized wave functions χ 1 (r 1 ), χ 2 (r 2 ) are described in Eqs. (9) and (10). We assume that the 1s wave functions are orthogonal to the 2s wave function: We calculate the ground state energy of a three-electron atom using variational principle.
We optimize the parameter ξ, defined in the wave functions χ 1 (r 1 ), χ 2 (r 2 ), χ 3 (r 3 ), and obtain the minimum value of the Hamiltonian operator H φ (ξ), which is defined as We divide the above Hamiltonian (31) into five parts, where is the kinetic energy of the three electrons, is the potential energy of the three electrons due to nuclear attraction, and are the interaction energies for inter-electronic repulsions in the system.
We minimize the Hamiltonian operator H φ (ξ) with respect to ξ, with such that and obtain ξ 0 = 0.697856, which put into Eq (37) gives the ground-state energy, 1 = −0.693979.

Infinite-dimension: D → ∞
At large-D-limit the effective ground state Hamiltonian for three-electron atoms, with correlation can be written as: where J(r 1 , r 2 , r 3 ) = 1 with γ ij = γ ij = cos θ ij , and θ ij is the angle between r i and r j . The quantities Γ (i) and Γ are called the Gramian determinants. In equation (39) the quantity Γ (i) Γ is effectively defined as: See page 111, equation (35) in 7 for more details.
We minimize the above effective-Hamiltonian with respect to the parameters r 1 , r 2 , r 3 , and θ 12 , θ 13 , θ 23 respectively and obtain the corresponding ground state energy ∞ = −0.795453.

First-order perturbations:
(1) D As the electrons reside in two orbits, 1s 2 2s, there are three electron-electron pairs: one 1 r 12 from 1s 2 , the two others 1 r 13 and 1 r 23 from 1s2s. Thus each (1) D coefficient is comprised from the three electron pairs: (1) (1) The D = 1 item is obtained via subsection 4.1. The D = 3 item is attained from Ref. 27 Here we will develop both D = 3 and D → ∞ bringing the third electron akin with the two-electron treatment in subsection 3.3. As the Hamiltonian is evident in equations (19) and (20), we start with the electronic wave function: The two normalized functions χ 1 (r 1 ), χ 2 (r 2 ) are taken care of in Eqs. (22), (23), (24) and (25). We assume that the 1s wave functions are orthogonal to the 2s wave function: The normalization is: with α = 3 2D . To obtain the first-order terms for D = 3 and D → ∞ we need to assemble some integrals associated with the key f (D) function shown in Eqs. (2) and (26). The output is: and the hyprgeometric function F 1 2 , 3−D 2 ; D 2 ; y enters in (26).
The parent integral is, and From G D (a, b) we compute the following integral: In the integrals, we used the normalized wave functions χ 1 (r 1 ), χ 2 (r 2 ), and χ 3 (r 3 ) already specified, such a typical term: From Eq. (53), we see that we have to put a = 2 and b = 1, so y = 1/9. In Eq. (48) (55) the hyprgeometric function is available in tabulations. 28 We computed up to D = 10 6 to see that the function converges to For D = 3, the function gives

Interpolation for D=3
Again we use the interpolation formula shown in Eq. (6), now with λ = 1/Z = 1/3. The input from our A, B, C subsections was: Our interpolation gave the Li atom ground-state energy with error 1%: 3 = 0.839648, compared with the exact result 3 = 0.830896. 26

Four-electron: Beryllium
The electronic structure of the beryllium atom is highly interesting because it's implication in different areas of modern science, for e.g. stellar astrophysics and plasmas, high-temperature physics etc. The ground-state energy for the Be-atom has been calculated by applying various methods for e.g. the Configuration Interaction (CI) method with Slater-type orbitals (STOs), 29 the Hylleraas method (Hy), 30 the Hylleraas-Configuration Interaction method (Hy-CI), 31 and the Exponential Correlated Gaussian (ECG) method. 32,33 In this section we present the dimensional interpolation formula, by using the results from D = 1 and D = ∞ limit, to obtain the ground state energy of the four-electron atoms.
With dimensional interpolation we obtain the ground state energy of beryllium atom to be 3 = −0.910325, compared to the exact energy 3 = −0.916709, with a percentage error of 0.6%.

One-dimension: D=1
In Four-electron atoms, with nuclear charge Z = 1/λ, the exact Hamiltonian in onedimension using atomic units can be written as: In beryllium atom we consider the first two electrons are in 1s states, and other two electrons are in 2s states with spatial part being symmetric (both electrons are in the same state) and the spin part in the antisymmetric state. We write spatial part of the electronic wave function as follows: φ(r 1 , r 2 , r 3 , r 4 ) = χ 1 (r 1 )χ 2 (r 2 )χ 3 (r 3 )χ 4 (r 4 ), The three normalized wave functions χ 1 (r 1 ), χ 2 (r 2 ), χ 3 (r 3 ) are described in Eqs. (9), (10) and (30). We assume that the 1s wave functions are orthogonal to the two 2s wave functions χ 3 (r 3 ) and χ 4 (r 4 ) = 9ξ 20 We calculate the ground state energy of a four-electron atom with variational principle.
We optimize the parameter ξ, defined in the wave functions χ 1 (r 1 ), χ 2 (r 2 ), χ 3 (r 3 ), χ 4 (r 4 ), and obtain the minimum value of the Hamiltonian operator H φ (ξ), which is defined as: We divide the above Hamiltonian into five parts, where is the kinetic energy of the four electrons, is the potential energy of the four electrons due to nuclear attraction, and are the interaction energies for inter-electronic repulsions in the system.

Infinite-dimension: D → ∞
In large-D-limit the effective ground state Hamiltonian for four-electron atoms, with inter-electronic correlation can be written as: where J(r 1 , r 2 , r 3 , r 4 ) = 4 i,j=1 1 with γ ij = γ ij = cos θ ij , and θ ij are the angle between r i and r j . The quantities Γ (i) and Γ are the Gramian determinants. In equation (70) the quantity Γ (i) Γ is effectively defined as follows: See page 111, equation (35) in 7 for more details.

First-order perturbations:
(1) D As the electrons reside in two orbits, 1s 2 2s 2 , there are six electron-electron pairs: one 1 r 12 from 1s 2 , four others 1 r 13 , 1 r 14 , 1 r 23 , 1 r 24 from 1s2s; and another lonely 1 r 34 from 2s 2 . Each (1) The D = 1 item is obtained via subsection 5.1. Here we will develop both D = 3 and D → ∞ bringing the fourth electron akin with the three-electron treatment in subsection 4.3. As the Hamiltonian is evident in equations (19) and (20), we start with the electronic wave function: The two normalized 1s wave functions χ 1 (r 1 ), χ 2 (r 2 ) are taken care of in Eqs. (22), (23), (24) and (25). We assume that the 1s wave functions are orthogonal to the 2s wave functions χ 2 (r 2 ), defined in 46, and : with normalization constant N 1 defined in 47.
We take same approach as subsection 4.3 to calculate the first-order term (the 2s 2 electron-electron repulsion term) at D → ∞ limit with the help of equations (50, 52): with, y = a−b a+b 2 and f (D) function shown in Eqs. (2) and (26). This is same functional expression as in lithium atom (53), but the arguments are different.
To calculate the first-order perturbation coefficient 1 r 34 for beryllium we use the normalized wave functions χ 1 (r 1 ), χ 2 (r 2 ), χ 3 (r 3 ) and χ 4 (r 4 ) already specified, which gives rise to a typical term like From the above Eq. (79), we see that we have to put a = 1 and b = 1 , so y = 0. In Eq.(78) the hyprgeometric function and f (D) → 2 −1/2 at D → ∞ limit.
At D → ∞ limit (78) gives 1 For D = 3 we use the following formula from 1 and: 28 At D = 3 the 2s wave function with α = 1 such that To calculate the inter-electronic repulsion energy 1 r 34 from (85) we use the above type of integrals G k 3 (a, b) in Eq. (82) and K 3 (i, j, k) in Eq. (83), with a = 1, b = 1, and k = 0.
With the help of (82, 83) we calculate the first-order coefficient (2s-2s part) for beryllium atom in three dimension:

Interpolation for D=3
We again use the interpolation formula shown in Eq. (6),

One-dimension: D=1
In H 2 , with nuclear charge of each atom Z, the electronic part of the Hamiltonian in one-dimension using atomic units can be written as: 20,39 with a = R/2, where R is the distance between the two nuclei located at r = ±a; also λ = 1/Z = 1. The Hamiltonian energy eigenvalues provide symmetric and antisymmetric states under exchange of the electrons. The symmetric state pertains to the ground-state potential energy: 20 The total binding energy is obtained by adding the nucleus-nucleus-interaction term (1/R) with the electronic energy.
with a = R/2 and J(ρ 1 , ρ 2 , z 1 , z 2 , φ) = 1 In the D → ∞ limit, the Hamiltonian has two locations for electrons, namely: symmetric, with ρ 1 = ρ 2 and z 1 = z 2 , and antisymmetric, with ρ 1 = ρ 2 and z 1 = −z 2 . When R has the nuclei well apart, in the symmetric case, both electrons cluster near one of the nuclei (H 2 → H − + H + ); in the antisymmetric case, each electron resides near just one of the nuclei (H 2 → H + H). Thus, the antisymmetric case is much more favorable for the ground-state energy.
We minimize the Hamiltonian (90) with respect to ρ's and z's to obtain the ground state energy, ∞ (R); we numerically evaluate the corresponding optimized parameters ρ * 1 , ρ * 2 , z * 1 , z * 2 , and φ * for different values of R.
The total binding energy is obtained by adding to ∞ (R) the internuclear-interaction term (1/R).

Interpolation for D=3
Unlike the atoms, our interpolation will be different for a molecule. An atom has only one nucleus, with the electrons orbiting about the positive charge; then our interpolation deals with the first-order perturbation works well but not for a molecule. For a diatomic molecule, V (R) is fundamental, with R distance roaming between the nuclei. As mentioned in Eqs. (4) and (5), our interpolation for H 2 uses a modified rescaling scheme developed by  with the D = 1 and D → ∞ dimensional limits: The rescaled distances are: In D = 1 : r i → r i /3 and R → R /3, for i = 1, 2 ; In D → ∞ : The rescaled Hamiltonians have distinct factors in the kinetic and potential energy parts: In D = 1: Hamiltonian (88) becomes: In D → ∞: Hamiltonian (90) becomes: with a = R/2 and We minimized these rescaled Hamiltonians (93) with respect to the rescaled distances (92).

Conclusion and prospects
The formula used for atoms we consider unorthodox, as it recently emerged 14 whereas other D-interpolations are elderly. 44,45 The fresh aspect links the energies 1 and ∞ together with the first-order perturbation coefficients Those perturbations arise from of electron-electron pair interactions, 1/r ij ; they actually provide much of the dimension dependence. For H 2 we used a different scaling than with the atoms, since H 2 links the distance R between the two nuclei. Then the rescaling is: R → 1/3R for D → 1; R → 2/3R for D → ∞. Interpolating between the dimensional limits gave a fair approximation of the binding energy for D = 3, when compared with the full configuration interaction (FCI).
In tally, our sections 3 4 5 treat He, Li, Be; in 6 dealt with H 2 . In subsections we describe the D = 1 limit, the D = ∞ limit, the first-order perturbations, and the interpolation output.
The ingredients of the interpolation are well suited for computing. We expect the method to hold true for larger atomic, molecular and extended systems. More than ground-state energies are accessible. However, there are prospects for combining dimensional limits to serve other many-body problems. One is examining dimensional dependence of quantum entanglement. 46,47 Another is the isomorphism between the Ising model 48 and two-level quantum mechanics. 49 Long ago the Ising model was solved in one, two and infinite dimensions, 50-52 as well much activity near four dimensions. 53 The unknown solution at D = 3 remains a challenge even by quantum computing. 54,55 More light on the solution might come by blending of dimensions akin to our unorthodox interpolated formula.