N-particle wave function

The number of density matrix elements 7" scales with m, i.e. it does not directly depend on n. However, since m should be choosen roughly proportional to n, there is a scaling n. If it were possible to take the 7p rather than the Cl coefficients as variational parameters, we would have got rid of the scaling problem of full Cl. Unfortunately the cannot be regarded as variational parameters, unless one can impose conditions which guarantee that a 7-matrix is derivable from an n-particle wave function. This n-representability problem has played a big role in the late sixties and has been most thoroughly been formulated by Coleman [88, 89]. Unfortunately a simple solution of the n-representability problem which allows one to replace the n-electron wave function by the two-particle density matrix in a variational approach has not been found. Note that 7 is not independent of 7", but can be derived from this by partial contraction... 

This functional is well defined for every normalized 1 ) in C, the space of all square integrable N-particle wave functions. In the above definition, 2( P0 is the same for all I F ) having the same density as I P). Thus, Q( ) changes only when the density changes. Further, (2( F) is not the kinetic plus interaction energy of I P)... 

The n-electron wave function r t) describes the motion of the electrons in the field of the nuclei. Due to the electron-electron interaction term in the Hamiltonian, this equation cannot be solved without approximations. The HF approximation assumes that the n-particle wave function r) can be written as an antisymmetrized product of one-electron functions lF (ri) (so-called orbitals) ... 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

On the other hand, more elaborate transformations of matter, typically chemical - especially biochemical - reactions, involve coordinated motions of several electrons and nuclei, far beyond the scope of the independent particle model. Their microscopic description and interpretation require us to develop new theoretical concepts and tools. Initial stages of this development have been in progress for a couple of decades, centering on the study of N-particle wave functions in their configuration space with 3N dimensions. 

The ground-state density pf (x) is related to the N-particle wave function ... 

This wave function is degenerate in the sense that the same energy is obtained whenever two particles are interchanged. A total of N different wave functions can be obtained in this way. Since the total wave function must be anti-symmetrical in the exchange of any two particles, it is correctly represented by the determinant... 

Now, if the many-body (electron) problem can be arranged in such a way that the many-body, nonseparable wave function is expressed in terms of a separable wave function, which depends on N single-particle wave functions (Hartree approximation), i.e.,... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

The basic variable in density functional theory (DFT)22 is the electron density n(r). In the usual implementation of DFT, the density is calculated from the occupied single-particle wave functions (r) of an auxiliary system of noninteracting electrons... 

We also discuss the generalization of density-functional theory to n-partical states, nDFT, and the possible extension of the local density approximation , nLDA. We will see there that the difficulty of describing the state of a system properly in terms of n-particle states presents no formal difficultie since DFT is directed only at the determination of the particle density rather than individual-particle wave functions. The extent to which practical applications of nDFT within a generalized Kohn-Sham scheme will provide a viable procedure is commented upon below. 

In order to generalize the HF equations to n-particle states, we perform a variational procedure. In this procedure, it is convenient to identify clearly and uniquely the particle space in which the two-particle wave functions are acting. This identification is considerably easier when the particle pairings are uniquely and distinctly defined, as is done in constructing the wave function or in partitioning the Hamiltonian. We can now obtain an eigenvalue-like... 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

The recent discovery of ceramic high-Tc superconductors has forced a re-examination of the basic concepts and physical assumptions employed in current theoretical approaches. In reexamining basic concepts, it is well to remember that the true N-electron wave function may be expanded in terms of components each of which is made up of N single particle functions and that this expansion can be made in (at least) two different ways ... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an N-electron wave function the operator alls) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

With the development of material science, fine chemistry, molecular biology and many branches of condensed-matter physics, the problem of how to deal with the quantum mechanics of many-particle systems formed by thousands of electrons and hundreds of nuclei has attained relevance. An alternative of ab-initio methods is the density functional approach [8-10] which gives results of an accuracy comparable to ab-initio methods. The density-functional method bypasses the calculation of the n-electron wave-function by using the electron density p r). The energy of a many-electron system is a unique function of electron density. The computational work grows like instead of in HF. 

The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly rather, its existence is inferred from the various experiments whose outcome is most rationally explained using a wave function interpretation of quantum mechanics. Further, the A -particle wave function is a rather complicated construction, depending on 3N spatial coordinates as well as N spin coordinates, correlated in a manner that almost defies description. By contrast, the electron density of an N-electron system is a much simpler quantity, described by three spatial coordinates and even accessible to experiment. In terms of the wave function, the electron density is expressed as... 

For the longitudinal vibrations of the diatomic chain with N unit cells, stands for (q, j = 1) and S2j for (q, j = 2). For the "single particle" wave functions we use the corresponding notation... 

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