The Many-Body Schrodinger Equation

The Schrodinger equation for a many-electron system is written as  

The simplest approach to approximating a solution to Eq. 4.3 is to assume that all the electrons move independently of one another. That is, imagine they mumally interact only via an averaged potential energy. This is known as the Hartree approximation. It enables us to write the Hamiltonian for the A -electron system as a sum of N one-electron Hamiltonians, and the many-body wave function as a product of N one-electron wave 

Write the general expression for the wave function for an A/-electron system using the Slater determinant. 

Using a single Slater determinant as a trial ground state wave function for the A-electron system, it is found upon application of the variational principle that the one-electron wave functions themselves must satisfy the following equation, which is provided here without derivation  

ij/j is the jth one-electron orbital accommodating the /rth electron with spatial coordinate and spin coordinate Likewise, the rth electron resides in the ith orbital denoted by (/r,. The Lagrange multiplier, Sj, guarantees that the solutions to this equation forms an orthonormal set and is the expectation value for the equation. Hence, it is the quantized one-electron orbital energy. The second term within the 

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We proceed now to describe some of the most common approximations to the defect environment and the many-body Schrodinger equation and some simple models relating to defects in semiconductors that have been deduced from them. 

Ah, the crux of the problem, is it not Up until now, we ve just assumed we have some set of molecular orbitals i or Vu which we can manipulate at will. But how does one come up with even approximate solutions to the many body Schrodinger equation without having to solve it Start with the celebrated linear combination of atomic orbitals to get molecular orbitals (LCAO-MO) approximation. This allows us to use some set of (approximate) atomic orbitals, the basis functions which we know and love, to expand the MOs in. In the most general terms,... 

In most present implementation of DFT, the many-body Schrodinger equation is replaced by a set of coupled effective one-particle equations, the so-called Kohn-Sham equations [21]... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

While these problems, starting from the many-body Schrodinger equation, and ranging to pattern formation in driven complex fluids and chemical reactions at the biomolecule-membrane interface in aqueous solution, clearly are not yet solved and will remain at the forefront of challenges for many years to come, it must be emphasized that steady and important progress towards reaching these goals can be anticipated. Thus, we can expect that the role of computer simulation as the central and basic methodic approach of condensed matter theory will become even much more important in the future. 

One would then wish to solve the Schrodinger equation for the wavefunction of the electrons in the potential of the nuclei. This wavefunction would define the state of the system and can be used to obtain all observable properties of the system. The problem is that the many-body Schrodinger equation cannot be solved accurately, even using the most powerful of computers, except for the simplest of systems, consisting of only a few electrons. 

Computational quantum mechanical methods, such as the Hartree-Fock method (Hehre et al., 1986 Szabo and Ostlund, 1989 Levine, 2000), were developed to convert the many-body Schrodinger equation into a singleelectron equation, which can then be solved tractably with modern computational power. The single-electron equation is an approach by which the state (or wavefunction) of each electron is computed within the field... 

Most band-structure calculations in solid-state physics are actually calculations of the KS eigenvalues q.39 This simplification has proved enormously successful, but when one uses it one must be aware of the fact that one is taking the auxiliary single-body equation (71) literally as an approximation to the many-body Schrodinger equation. DFT, practiced in this mode, is not a rigorous many-body theory anymore, but a mean-held theory (albeit one with a very sophisticated mean held i>s(r)). 

Much progress has been made in other areas of quantum physics earlier in the absence of complete solutions of the many-body Schrodinger equation. In our view, until recently the lack of direct experimental information and direct unambiguous computational results have conspired with the high dimensionality and richness of mechanism in the IVR problem to inhibit the development of simple pictures of quantum intramolecular vibrational energy flow. 

In principle, the formulation of the theory is straightforward. One wishes to solve the many-body Schrodinger equation... 

The multiple scattering optical potential formalism of Watson [Wa 53] provides a formal solution of the nonrelativistic many-body Scluodinger equation for the projectile-nucleus system in terms of a systematic expansion in elemental, many-body operators defined in ref. [Wa53]. The many-body Schrodinger equation is... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

The first step of any DFT is the proof of a Hohenberg-Kohn type theorem [6]. In its traditional form, this theorem demonstrates that there exists a one-to-one correspondence between the external potential and the (one-body) density. The first implication is clear With the external potential it is alwaj possible (in principle) to solve the many-body Schrodinger equation to obtain the many-body wave-function. From the wave-function we can trivially obtain the density. The second implication, i.e. that the knowledge of the density is sufficient to obtain the external potential, is much harder to prove. In their seminal paper, Hohenberg and Kohn used the variational principle to obtain a proof by reductio ad ahsurdum. Unfortunately, their method cannot be easily generalized to arbitrary DFTs. The Hohenberg-Kohn theorem is a very strong statement From the density, a simple property of the quantum mechanical system, it is possible to obtain the external potential and therefore the many-body wave-function. The wave-function, in turn, determines every observable of the system. This implies that every observable can be written as a junctional of the density. 

The basic concept is that instead of dealing with the many-body Schrodinger equation, Fq. (2.1), which involves the many-body wavefunction P ( r ), one deals with a formulation of the problem that involves the total density of electrons (r). This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 

However, solving the many-body Schrodinger equation is much easier said than done, and it is impossible to solve exactly for anything but the simplest model systems with 1, 2, or perhaps 3 electrons, or an infinite jeUium system, the homogeneous electron gas (see Section 2.4.2). Since solids contain lots ( 10 ) of electrons and the potential due to the nuclei is far from the constant of jehum, we have a challenge. The root of the problem is well known. It is the quantity V , the electron-electron interaction, which contains all the many-body physics of the electronic structure problem. It depends on (at least) 3N spatial coordinates, which are all coupled by the operator... 

Equations (61 )-(63) are the celebrated Kohn-Sham equations. They replace the problem of minimizing E[ri by that of solving a single-body Schrodinger equation. (Recall that the minimization of E[n originally replaced the problem of solving the many-body Schrodinger equation )... 

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