## Abstract

Prediction of properties of solids (semiconductors) is based almost entirely on the first-principles methods. The first principles theories are far from being perfect and new schemes are developing. In this study, we do not follow the traditional one-particle-in-effective-field concept. Instead, all Coulomb interactions between particles are treated in their original form, i.e., particle-particle discrete interactions. Two-particles Coulomb excitations theory in a crystal lattice is proposed, along with a method for calculations of physical measurables. Most important, the relevant particles are not electrons but pseudo-electrons with both the Coulomb interaction mode and the effective mass different from those of electrons. The unitary transformation represents the many-body system as an ensemble of two-pseudo-electron excitations without neglection of the terms in a Hamiltonian. The many-particle wave function, being derived in a non-trivial two-particle form, ensures a full description of exchange-correlation and screening effects, for both ground and excited states. As an example, the energy of a many-electron system and the quasiparticle energies are expressed in an elegant integral closed-form and compared with the Density Functional Theory. The proposed scheme possibly opens a new route toward the numerical evaluation of properties of many-particle systems.

### Keywords

- solid state
- quantum mechanics
- many-electron problem
- first principles
- properties of semiconductors

## 1. Introduction

Since the creation of Quantum Mechanics (QM), the theories for the description of many-electron systems, with

Despite differences, all first-principles theories have a common foundation—the concept of * one-particle in an effective-field* concept. This concept leads to the description of a many-particle system in terms of one-electron wave functions,

*between couples of electrons (and the interactions of higher order are neglected), these schemes are called two-particle methods. Consequently, only the approximation solutions are possible and different sophisticated methods are utilized to increase the accuracy [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].*exclusively

In this study, we look at the many-body problem from a very different angle. First, we try to find the answer to the following question: Is it possible to reduce the many-electron problem to the * exactly two-particle* scenario when each particle interacts by a Coulomb force solely with one another particle? The motto here is the main principles of Quantum Mechanics which always brings into focus the most elementary levels of any effect. Obviously, in terms of interactions between particles, the interaction between

*particles is an elementary interaction, and all other interactions, are superpositions of two-particle interactions. Hence, one may ask if it is possible to canonically transform the many-body Hamiltonian,*two

*two-particle interactions, and all other excitation orders are not presented?*exclusively

If this question is positively answered, the relevant particles in canonically transformed Hamiltonian obviously * are not electrons* and should have different properties. Here, we demonstrate that such canonical transformation,

*that adequately represent the canonically—transformed many-body system. The two-particle excitations describe the mutual scattering of coupled pseudo-particles in a periodic crystal field that obeys the Schrodinger equation. Note, that these excitations are different from both quasi-electrons and excitons which describe the interaction between an excited electron and a hole.*pseudo-electrons

In other words, we show here that many-electron problems can be * exactly* expressed as an ensemble of two-particle excitations, without any truncation or omissions in Hamiltonian. Since the two-particle problem was extensively studied in many QM treatments and can be solved by different methods [14], our approach promises an elegant solution for the many-electron problem. Strangely, in a sea of literature on the many-body problem, we could not find analogous canonical transformation. All two-particle theories in the many-body problem, e.c. double cluster methods [10, 11, 12, 13] or DFT [5, 6, 7, 8], consider

*in a truncated Hamiltonian with omitted high order electronic interactions. Another example is the*electrons

*in Quantum Chemistry [15, 16] or two-particle coordinate (spin) wave functions which represent a generalization of one-electron orbitals accounting for intra-orbital correlation effects. Again, the geminal theories assume that the relevant particles are*geminals

*and not pseudo-electrons. Note, that our pseudo-electrons are*electrons

*from electrons in terms of their properties, which are their effective mass and the Coulomb mode of interaction. Moreover, the total number of pseudo-electrons,*very different

Consequently, we believe that our approach is very different from existing many-body theories. The numerical justification of our scheme requires special extensive work and is surely out of a scope of a current study. Meantime, we hope to provide such numerical validations later. The theory utilizes the following methods: special unitary trans-formation of many-electron Hamiltonian,

## 2. Unitary transformation of a Hamiltonian

We consider the periodic crystal lattice with cyclic boundary conditions, which has totally * or any excited* state of our interest. Each single-electronic state, with an index,

where the last term describes the Coulomb interactions between electrons. For simplicity, we omit here the spin-orbital and other high-order interactions. To derive two-electron Coulomb excitations, one needs to apply to

where

The unitary transformation,

One can see, from Eq. (5), that the Hamiltonians

with both the renormalized mass of electrons, * pseudo-electrons*, and the renormalized pseudo-potential as an

*Each original electron, with index*effective pseudo-potential.

*,*different index

*possible electron–electron Coulomb interactions reside solely*all

*the two-particle excitations (Figure 1). Since there are no Coulomb interactions*within

*the excitations, we can treat each two-particle excitation as a*between

*with its two-particle wave function,*quasi-closed subsystem,

## 3. Fermionic many-electron wave function in terms of two-particle excitations

The unitary transformation,

yields the transformed Schrodinger equation,

This unitary transformation has the following invariants: 1. The unitary transformation,

This invariant follows from the definition of pseudo-electrons: the fermionic anti-symmetry assumes that two pseudo-electrons of interest belong to different groups,

4. The unitary transformation does not change the many-particle wave function,

Eq. (12) is an exact extension of a many-electron wave function in a * finite series* of two-particle wave functions. The derivation of Eq. (12) is based on a combinatorics of a Slater determinant with suppressed interactions, combined with the substitution

## 4. Two-particle wave functions

The Hamiltonian of two pseudo-electrons in an effective potential,

The two-electron wave functions,

It is known that the two-electron problem can be solved by different methods, see Ref. [22], and here two-particle Green function method is utilized. In this method, one needs first to suppress the Coulomb interaction between the electrons, or the last term in the right part of Eq. (13). As a result, the Hamiltonian becomes a sum of two one-particle Hamiltonians:

The one-particle Green function,

where

The calculation of an integral in the right part of Eq. (17), with the help of Eq. (16), provides the following result:

Next, in an accordance with the Green function method, in Eq. (13), we should turn on the Coulomb interactions,

Hence, the corrections,

The integral Eq. (21) can be solved by different methods, e.c. by the iteration method:

This procedure, after

## 5. Energy functional and quasi-particle energies

The quasi-particle energy,

Here

The contributions to the

The expressions (25)–(29) are reduced, after considerable analytical efforts, to the closed integral forms:

where:

As we mentioned above, the quasi-particle energies,

## 6. Discussion and conclusions

The many-electron problem has a unique exact analytical solution [Eq. (12)], or many-electron wave function, * any magnitude* of interactions between particles, and no higher-order excitations are required to be included in the scheme. Moreover, one can see from Eq. (4) that, in a unitary transformed lattice, the electronic excitations with an order larger than two simply do not exist. Generally, a developed scheme is a natural consequence of the fact that any elementary interaction includes two and only two particles. It is worth remembering that the two-particle wave functions,

*the original electrons. This point is an essential novelty of our theory. Indeed, two-electron wave functions are presented in other many-body theories [5, 6, 7, 8, 10, 11, 12, 13], but these representations always assume truncations or neglect the terms in a Hamiltonian. In an analytical part of our calculations, the tremendous help comes from the specific form of Eq. (12) for the many-body wave function,*not

It would be instructive to compare our theory with the DFT scheme. First, the original DFT method was developed for ground states only, and our approach works universally for * any state*, no matter ground or excited. The Eqs. (30)–(31), for both the total energy,

*, the result we could not find in literature. The differences between DFT and our approach are laid within the most delicate parts of the energy functional, i.e., the screening correction,*ground and excited states

*of a many-body system, due to the relaxation of the atomic orbitals and the screening. Note, that the problem of a response is*response

*due to one-particle-in-effective-field concept utilized in these theories. This response is usually described by means of the dielectric matrix,*entirely

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