Antisymmetrized wave function Slater determinant

John C. Slater (1901-1976), American physicist, for 30 years a professor and dean at the Physics Department of the Massachusetts Institute of Technology, then at the University of Florida, Gainesville, where he participated in the Quantum Theory Project. His youth was in the stormy period of the intense development of quantum mechanics, and he participated vividly in it. For example, in 1926-1932, he published articles on the ground state of the helium atom, the screening constants (Slater orbitals), the antisymmetrization of the wave function (Slater determinant), and the algorithm for calculating the integrals (the Slater-Condon rules). 

Every normalizable antisymmetric wave function can be expressed as the sum of a series of Slater determinants built up from a complete basic set of one-electron functions. 

Such a wave function is known as a Slater determinant. In general, when we deal with antisymmetrized wave functions, we use a compact notation for the Slater determinant ... 

It is evident that Lennard-Jones was following the track opened by Lewis, by concentrating on the pair of electrons. To get some insight into P2(xi, x2), it is natural to start with the simplest antisymmetric wave function, a Slater determinant constructed by real orbitals. In this case, one obtains... 

The zeroth-order antisymmetric wave functions of closed-subshell states, states that have only one electron outside a closed-subshell configuration, and states that are one electron short of having a closed-subshell configuration can be expressed as a single Slater determinant [e.g. (1.259)]. However, for open-subshell states in general, one has to take an appropriate linear combination of a few Slater determinants to get a state that is an eigenfunction of L2 and S2. 

In the MO approach appropriate to outer s and p electrons, the simple formalism does not distinguish between a covalent-ionic band and a metallic band. The use of determinantal (antisymmetrized) wave functions automatically introduces correlations between electrons of parallel spin. Traditionally the many-electron wave function has, at best, been represented by a single Slater determinant of one-electron wave functions (Hartree-Fock approximation), whereas the true wave function would be given by a series of such determi-... 

Finally, it should be mentioned that the structures outlined above are only used for convenience. In the total antisymmetrized wave-function (a Slater determinant, section III) the distinction between different states of hybridization disappeare. Thus by proper transformation (keeping the total wavefunction unchanged) we can replace the two or-two ttd molecular orbitals of N3 by four equivalent bananashaped orbitals (see section III.F.l). (In the latter picture the central nitrogen atom may be regarded as being in a hybridization state which resembles rp .) This transformation is similar to that employed for the nitrogen and acetylene molecules, where the three ott bonds can be replaced by three equivalent bonds . 

The antisymmetric wave function, shown in Equation 28-SI. is a more compact way of writing a Slater determinant (Eq. 28-53). In a Slater determinant, an exchange of any two rows or columns results in the same wave function multiplied by -1. This is another statement of the Pauli exelusion principle. The columns in Equation 28-53 are the single electron wave functions. Equation 28-53. however, is only an approximation, since the electrons are independent of one another and therefore not eorrelated. This eorrelation problem reveals itself when ealculating the energies and is di.s-cussed below. 

The simplest antisymmetric wave function, which can be used to describe the ground state of an N-electron system, is a single Slater determinant,... 

The most suitable many-electron molecular wave functions are the antisymmetrized products (Slater s determinants) of mono-electronic wave functions, and it may be shown that any antisymmetric wave function T = T (a i, x, , x ) may always be expanded in a series of such determinants ... 

The simplest wave function to describe a many-electron system is a Slater determinant built by orthogonal one-electron wave functions. Electrons are fermions and accordingly they have to be described by an antisymmetric wave function. For an N-electron system, the Slater determinant has the form ... 

There is another notation that can be used to write the antisymmetrized wave function of Eq. (18.6-5). It is known as Slater determinant. Adetermhiaiitis a quantity derived from a square matrix by a certain set of multiplications, additions, and subtractions. There is a brief introduction to matrices and determinants in Appendix B. If the elements of the matrix are constants, the determinant is equal to a single constant. If the elements of the matrix are orbitals, the determinant of that matrix is a single function of all of the coordinates on which the orbitals depend. The wave function of Eq. (18.6-5) is equal to the determinant ... 

For the correlation energy no general explicit expression is known, neither in terms of orbitals nor in terms of densities. A simple way to understand the origin of correlation is to recall that the Hartree energy is obtained in a variational calculation in which the many-body wave function is approximated as a product of single-particle orbitals. Use of an antisymmetrized product (a Slater determinant) produces the Hartree energy and the exchange... 

As discussed above, it is impossible to solve equation (1-13) by searching through all acceptable N-electron wave functions. We need to define a suitable subset, which offers a physically reasonable approximation to the exact wave function without being unmanageable in practice. In the Hartree-Fock scheme the simplest, yet physically sound approximation to the complicated many-electron wave function is utilized. It consists of approximating the N-electron wave function by an antisymmetrized product4 of N one-electron wave functions (x ). This product is usually referred to as a Slater determinant, OSD ... 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

Interchange of two rows of the Slater determinant changes the sign of the wave function, which is therefore antisymmetric with respect to interchange of electrons. When two rows or columns are identical, the determinant is zero. The Slater determinant wave function therefore obeys the Pauli exclusion principle for fermions. 

Every fermionic wave function has to be antisymmetric when exchanging the coordinates of any two particles (Pauli principle) which is not fulfilled for a simple product ansatz as in Eq. (3). Therefore, we explicitly have to antisymmetrize the Hartree product THP and obtain the Slater determinant ... 

Recall that the minimum requirement for a many-electron wave function is that it be written as a suitably antisymmetrized sum of products of one-electron wave functions, that is, as a Slater determinant of MOs [see equation (A.68)] In Chapter 2 and Appendix A, we find that the condition that this be the best possible wave function of this form is that the MOs be eigenfunctions of a one-electron operator, the Fock operator [recall equation (A.42)], from which one can choose the appropriate number of the lowest energy. The Fock operator in restricted form, F( 1) [RHF, the UHF form was given in equation (A.41)], is given by... 

In the Hartree-Fock method, the molecular (or atomic) electronic wave function is approximated by an antisymmetrized product (Slater determinant) of spin-orbitals each spin-orbital is the product of a spatial orbital and a spin function (a or ft). Solution of the Hartree-Fock equations (given below) yields the orbitals that minimize the variational integral. Thus the Hartree-Fock wave function is the best possible electronic wave function in which each electron is assigned to a spatial orbital. For a closed-subshell state of an -electron molecule, minimization... 

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