Open access peer-reviewed chapter

Many-Electron Problem in an Atomic Lattice Reduced Exactly to Two-Particle Pseudo-Electron Excitations: Key to Alternative First-Principles Methods

Written By

Adil-Gerai Kussow

Submitted: 18 January 2022 Reviewed: 04 February 2022 Published: 13 April 2022

DOI: 10.5772/intechopen.103045

From the Edited Volume

New Advances in Semiconductors

Edited by Alberto Adriano Cavalheiro

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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.


  • 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 N electrons, have been developed for decades [1, 2]. The most elaborated applications of many-body theories are the first-principles calculations which are used virtually in all fields of modern physics [3]. These calculations became a tool that provides priceless information to describe both the undisturbed atomic configuration and the response of the system to an external perturbation. In terms of formalisms, the many-electron theories belong to several groups: 1. Perturbation-based theories [4]. These methods utilize a small α parameter expansion, which, often, is the ratio of the electron–electron Coulomb interaction energy to the total energy, α=Hint/H0. 2. Green function-oriented theories are generally more powerful than perturbation schemes, since they are not necessarily based on a small parameter. 3. Density functional-based schemes (DFT) [5, 6, 7, 8] explore a semiclassical concept of an electron gas [9]. 4. The coupled cluster theories [10] find solutions of the Schrodinger equation for many-electron problem without assumptions of DFT theory (e.c. electrons gas approximation). The goal of the cluster theories is the calculation of the wave function for the many-body system, Ψr1r2.rN, which depend on N coordinates, ri, of electrons. To treat the exchange-correlation effects, the coupled cluster techniques utilize the antisymmetric Slater determinant, D. The elements of D are the one-electron wave functions, φirit, derived within the effective-field approximation. The wave functions, φirit, are calculated assuming two or three electron excitations (Coulomb interactions between electrons), and D provides the many-electron wave function, Ψr1r2.rN. Different methods [10, 11, 12, 13], which include the parametrization and variational principles, allow to derive the ground and the excited states of the system. The coupled cluster approaches often demonstrate the superior accuracy of calculations if compare with DFT schemes, and these methods were successfully applied in different fields ranging from Chemistry to Nuclear Physics [10].

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, φirit, which are wave functions of free electrons, corrected due to the presence of other particles. The main difficulty here is that all N electrons interact with each other, with a large number of interactions NN1/2>>10. Consequently, the first-principles schemes consider a restricted number of Coulomb interactions between the electrons only, and the rest of the interactions are omitted. The numeric schemes usually select combinations of 2–4 electrons, and the higher interaction orders are ignored. If Coulomb interactions are considered exclusively 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].

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 two 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, Ĥ, to a form that involves exclusively two-particle interactions, and all other excitation orders are not presented?

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, T˜, exists. Consequently, we introduce new particles or pseudo-electrons 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.

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 electrons in a truncated Hamiltonian with omitted high order electronic interactions. Another example is the geminals 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 electrons and not pseudo-electrons. Note, that our pseudo-electrons are very different 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, NN1, is different from the total number, N, of electrons.

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, ĤĤ˜, two-particle Green function formalism for the calculation of the two-particle excitations, and combinatorics applied to Slater determinant for the derivation of a fermionic many-body wave function, Ψr1r2.rN. All parts of the approach are within traditional QM formalism [17, 18], and no other supporting approximations were utilized. We have found out, that our efforts are extremely rewarding in terms of results. They lead to general expression, in a closed form, for the main object of QM—many-electron fermionic wave function, Ψr1r2.rN. Just as an example, we derive from Ψ the electronic lattice energy functional which is the source for the calculation of quasi-particles (QP) energies, band structures, and optical spectra. These spectra are often obtained within Hedin’s GW approach [19] based on one-body single-particle Green’s function formalism. The QP spectra are required to obtain the optical spectra, which are often calculated from the Rohlfing-Louie electron–hole excitation model [20, 21], or the many-body perturbation Green function method. In a contrast, our approach allows us to calculate QP spectra, and hence, the optical spectra as well, based solely on the many-electron fermionic wave function, Ψ.

2. Unitary transformation of a Hamiltonian

We consider the periodic crystal lattice with cyclic boundary conditions, which has totally N0 electrons per unit cell. We are interested in quantum states of N<N0 electrons of a lattice, assuming, that the rest of N0N electrons, and core ions, can be replaced by an effective space-dependent pseudo-potential Vextr. Note, that the all-electron problem, N=N0, when the pseudo-potential, VextVION, is solely due to atomic nuclei, is covered within the same theory. The pseudo-potential, Vextr, generates the set, i, of the single-electron states, with indexes, i and the wave vector, ki. Some of the states are occupied with N electrons of our interest, having coordinates, ri and spin, σi. Each singe-electronic state can be occupied by two electrons, with opposite spins, by one electron, or is an empty state. The initial electronic state of a crystal is a set, i, of single-electronic states of our choice, with suppressed Coulomb interactions within the group of N electrons. The single-electronic states, with an index i and a wave vector ki, are described by one-electron ortho-normalized wave functions, φiki0ri, which are supposed to be known or previously calculated (e.c. these states can have the eigenfunctions for the pseudo-potential widely used in first-principles calculations). Note, that the initial electronic state of a many-electron system can be a ground state or any excited state of our interest. Each single-electronic state, with an index, s, if double occupied with electrons having opposite spins, splits into two degenerated electronic states. Next, based on the initial electronic state of a crystal, we turn on the Coulomb interactions between N electrons. Our goal is a calculation of the actual quantum states of the lattice, or the many-particle wave function of the system, Ψ, with all electron–electron Coulomb interactions included. This formulation of a many-body problem is a traditional one in first-principles calculation schemes and analytical many-body theories. The nonrelativistic Hamiltonian, Ĥ, for the many-electron problem is given by:


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 Ĥ a special unitary transformation. One can see that, in Hamiltonian (1), the sums are running over both one integer index (first and second terms) and two integer indexes (third term). This inconsistency or imparity in the number of indexes can be fixed with help of algebraic equality:


where ai is a member of a set (for example, if N=3, then Eq. (2) reads:


The unitary transformation, U+=U1, of the Hamiltonian, Ĥ, is equivalent to re-grouping terms of Ĥ, based on equality (2). This transformation, being applied to one-indexed terms of Ĥ, makes all terms of the Hamiltonian consistent in the number of indexes, which are now two. As a result, the transformed Hamiltonian Ĥ˜=UĤU1 becomes the sum of two-particle Hamiltonians, Ĥ˜ilrirl, which describe the two-particle excitations, with particle indexes, il>i:


One can see, from Eq. (5), that the Hamiltonians Ĥilrirl can be written as:


with both the renormalized mass of electrons, m˜N1m, and the renormalized effective potential, V˜extrVextr/N1. This renormalization, does not affect Coulomb interactions, Hintil=e2/rirl, and e˜=e. In the following text we will call the electrons with renormalized mass as pseudo-electrons, and the renormalized pseudo-potential as an effective pseudo-potential. Each original electron, with index s, in an effective pseudo-potential, V˜extrs, on energy level, Es, after the unitary transformation, splits into a group of N1 pseudo-electrons, each having the same space-spin coordinates, rsσs. The general coordinates of the pseudo-electrons, ξS=rSσS, include both the space coordinates, rS, and the spin coordinates, σS. The total number of pseudo-electrons, 2CN2=N2N, is larger than N, and each pseudo-electron interacts with solely one pseudo-electron from the other group. Figure 1, or an example of four electron system, illustrates these points more clearly. Note, that the pseudo-electrons, having the same index s, interact only with pseudo-electrons having the different index, s', with no interactions between pseudo-electrons within the same group. In a transformed lattice, the system of N electrons in an effective pseudo-potential, Vextr/N1, is an ensemble of NN1/2 two-particle excitations. Each two-particle excitation, described by two-electron Hamiltonian, Ĥilrirl, consists of two pseudo-electrons, in an effective pseudo-potential field, V˜extr, plus the Coulomb interaction between these two pseudo-electrons. Since the indexes il run from 1 to N, all Coulomb excitations are considered. Moreover, in a transformed lattice, all possible electron–electron Coulomb interactions reside solely within the two-particle excitations (Figure 1). Since there are no Coulomb interactions between the excitations, we can treat each two-particle excitation as a quasi-closed subsystem, with its two-particle wave function, ϕilrirl. Consequently, in a transformed system, the two-particle representation is self-sufficient, with no room for a single-electron effective-field concept. Since all Coulomb interactions are included in this scheme, any additional corrections would be a double-counting mistake. Despite the lack of Coulomb interactions between the two-particle excitations, they are still coupled by a different mechanism. Indeed, since each original electron, with an index S and the space-spin coordinate, rsσs, is presented simultaneously in N1 two-particle excitations, in a form of pseudo-electrons, the common space-spin coordinates couple excitations. This coupling forbids writing down the many-electron wave function, Ψ, as a product of two-electron wave functions, ϕilrirl, and a different method is required to derive Ψ.

Figure 1.

Two-electron excitations in canonically transformed Hamiltonian (example: 4 electron system). (a) Original 4 electrons, with indexes 1234 and space-spin coordinates r1σ1, r2σ2, r3σ3, r4σ4. Each electron has its own colour and 6 interactions between electrons are shown as black lines. (b) In canonically transformed Hamiltonian, each original electron splits into 3 pseudo-electrons which belong to the same group, and all pseudo-electrons have effective mass different from the effective mass of original electrons. All 3 electrons within each group, are described by the same spin-coordinates, group 1: r1σ1, group 2: r2σ2, group 3: r3σ3, group 4: r4σ4. (c) Within each group, there is no interactions between pseudo-electrons. Each pseudo-electron interacts solely with only one pseudo-electron from different group. There are 6 two-electron excitations. Two-electron excitation is a couple of pseudo-electrons from different group plus an interaction between pseudo-electrons (shown as black lines). The two-electron excitations are shown as 6 rounded rectangles, along with their two-electron wave functions, ϕ12r1r2,ϕ13r1r3,ϕ14r1r4,ϕ34r3r4,ϕ32r3r2,ϕ42r4r2.

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

The unitary transformation, U, being applied to the Schrodinger equation,


yields the transformed Schrodinger equation,


This unitary transformation has the following invariants: 1. The unitary transformation, E˜=UEU1=E, does not affect the energy of a system. This is an obvious invariant since re-grouping of the terms in Ĥ should not change the eigenvalues of Ĥ. 2. The unitary transformed Ψ˜k1k2.kNξ1ξ2.ξN is a fermionic antisymmetric wave function, with respect to space-spin coordinates of a pseudo-electrons:


This invariant follows from the definition of pseudo-electrons: the fermionic anti-symmetry assumes that two pseudo-electrons of interest belong to different groups, i,l. 3. The unitary transformation does not affect the one-particle (free particle) wave functions, φsks0ξm=φ˜sks0ξm, calculated, by definition, with no Coulomb interactions between electrons. Indeed, the transformation reduces both the kinetic energy operator and the potential energy operator by the same factor, 1/N1: Ĥφsks0=Enφsks0;Ĥ/N1φ˜sks0=Es/N1φ˜sks0, and, hence,


4. The unitary transformation does not change the many-particle wave function, Ψ˜=Ψ, and expresses it in a different form. Indeed, since all N1 pseudo-electrons, within the same group, have the same space-spin coordinates, the set, ξ1ξ2.ξN, of the coordinates of the electrons, coincides with the set of the coordinates of quasi-electrons, Ψ˜k1k2.kNξ1ξ2.ξN. The QM defines the wave function as a probability, P, to find ith particle at coordinate ξi, within phase volume, dξi The original electron, with index i, has the same probability P with any of pseudo-electron within its group, because they have the same coordinate, ξi. Consequently, the unitary transformed wave function being expressed in terms of pseudo-electrons is an invariant: Ψ˜=Ψ. Since a many-particle system is an ensemble of two-particle excitations, Ψk1k2.kNξ1ξ2.ξN can be expressed in terms of two-electron wave functions, ϕikilklξkξm, km,k<m. One can prove from combinatorics of a Slater determinant that Ψk1k2.kNξ1ξ2.ξN is a linear combination of two-particle wave functions, ϕikilklξkξm, with coefficients, which are products of, φsks0ξs, by Levi-Civita symbols, ei1,i2,iN:


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 ϕikilklξkξm instead of ϕ0ikilklξkξm.

4. Two-particle wave functions

The Hamiltonian of two pseudo-electrons in an effective potential, V˜extri+V˜extrl, which interact by means of the Coulomb force is given by:


The two-electron wave functions, ϕikilklrirl, for either ortho or para spin configurations, are the solutions of the Schrodinger equation:


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, GErr', is built from the single-particle wave functions, φnk0r:


where δ=0+, and the sums are taken over all single-particle electronic states, nk and all wave vectors, k. The two-particle Green function, GEr1r2r1'r2', is calculated from the one-particle Green function as an integral over the real axis, ε, of a complex plane energy plane, E:


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, Hintil=e2/rirl, and calculate the corrected two-electron wave functions, ϕilrirl. In symbol, ϕilrirl, for simplicity, we have omitted the k vector and ϕilrirl read as ϕikilklrirl. Based on the Green function method, the corrected two-electron wave functions, ϕilrirl, satisfy the following integral equation:


Hence, the corrections, Δϕilrirl, to undisturbed two-particle wave functions obey the equation:


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


This procedure, after n iterations, yields ΔϕilrirlΔϕnilrirl, a desirable accuracy of n1 order with respect to the Coulomb interaction, Hintil.

5. Energy functional and quasi-particle energies

The quasi-particle energy, EiQPki, is either the energy of the electron, P=E, or the energy of the hole, P=H, in ith state, with the wave vector, ki. EiQPki, can be obtained from the expression for the electronic part of the energy of a lattice, ET (energy functional):


Here NORM is the normalization, and ΔV0 is the volume of the elementary cell. The substitution of Eq. (12) into Eq. (23) yields ET as a sum of five contributions:


The contributions to the ET are associated with the following interactions: E0 is the total energy of the electrons in an effective potential, with the suppressed Coulomb interactions between electrons. Eee is the energy of the Coulomb interactions between electrons, Eee, with no screening. E0Scr is the correction to the E0 due to the screening. EeeScr is the correction to the Eee due to the screening. ESE is the exchange-correlation energy, or the so-called self-energy, ΣSE. The expressions for these contributions are mentioned below:


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




As we mentioned above, the quasi-particle energies, EiQPki, can be obtained from Eqs. (30)(39) as a sum of terms of their right parts with a fixed index i.

6. Discussion and conclusions

The many-electron problem has a unique exact analytical solution [Eq. (12)], or many-electron wave function, Ψ, which is a linear combination, with a finite number of terms, of two-particle wave functions. This result stands for 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, ϕil, in Eq. (12), describe the pseudo-electrons in a unitary transformed lattice, and not 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, Ψ. In Eq. (12), the coefficients, Kilkm, in front of the two-particle wave function, ϕil, are the products of the ortho-normalized non-disturbed (free) one-particle wave functions, φsks0. Consequently, in the calculation of a physical measurable, P=ΨP̂Ψdq (energy functional), many integrals of products, φsks0φsks'0 are zeros. As a result, the analytical expressions for the integrals can be relatively easily derived. In this chapter, just as an example, we choose a physical measurable P as an energy of a crystal, and express this energy in a closed form, in terms of 6D space-coordinates integrals. Since the wave function, Ψ, is known, a similar procedure can be carried out for any physical measurable of interest, such as momentum, impulse, currents, and conductivity…

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, E0, of the electrons with the suppressed Coulomb interactions between the electrons, and for the energy of the Coulomb interactions between electrons, Eee, with no screening, are identical to the calculation of DFT scheme [5, 6, 7, 8]. Since our result stands for all states, no matter ground or excited, the DFT theory exactly describes the energy, Eee, for both ground and excited states, 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, E0Scr, [Eq. (32)], the screening correction, EeeScr, to the Eee, [Eq. (33)], and the exchange-correlation energy, ESE, or the so-called self-energy, ΣSE, [Eqs. (34)(39)]. It is known that the calculation of the exchange-correlation energy is the most difficult problem in first-principles calculations [23]. In our scheme, the exchange-correlation energy appears as a unique closed integral form, without any additional approximations and efforts, which are required, for example, in Hedin’s calculations for the exchange-correlation energy [19]. Moreover, in a proposed scheme, in contrast to DFT and Hedin GW [19], both ground state and the excited states are treated on the same footing, within the same QM scheme. Additionally, since in our theory, all Coulomb interactions are treated on the same footing, we do not separate the “direct” Coulomb interactions from the “screened” Coulomb interactions. Still, in Eqs. (24)(29) for the total crystal energy, the screened, the direct, and the exchange-correlation interactions appear as profoundly separate contributions. There are two additional advantages of our approach in comparison with the Density Functional schemes [5, 6, 7, 8, 9] and other methods based on an effective-field concept [10, 11, 12, 13]. In these schemes, the band structure and other physical properties are calculated by adding or removing an electron, or a hole to the system. This procedure inevitably causes the problem of a response of a many-body system, due to the relaxation of the atomic orbitals and the screening. Note, that the problem of a response is entirely due to one-particle-in-effective-field concept utilized in these theories. This response is usually described by means of the dielectric matrix, ε̂, which is to be calculated self-consistently [3]. Consequently, the singe-particle nature of these schemes brings additional difficulties in defining the consistent quasi-macroscopic dielectric function. Since in our method we do not rely on the single-particle effective-field model, the problem of response simply does not exist, with no need for the dielectric relaxation matrix, ε̂. Moreover, in a suggested method, the many-electron wave function provides a self-sufficient input for calculation of any measurable, with no need for inclusion of hole–electron interactions which would be a double-counting mistake.


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Written By

Adil-Gerai Kussow

Submitted: 18 January 2022 Reviewed: 04 February 2022 Published: 13 April 2022