Antisymmetric many-electron wave function

At small R on the other hand, the antisymmetrized, many-electron wave function... 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

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 ... 

The requirement that electrons (and fermions in general) have antisymmetric many-particle wave functions is called the Pauli principle, which can be stated as follows ... 

A many-electron wave function must be antisymmetric... 

Any many-electron wave function must be antisymmetric to the interchange of the spacial coordinates and spin (collectively referred to as the vector q) of any pair of electrons i and /. 

The overall many-electron wave function is formed from the Hartree-Fock orbitals as an antisymmetrized product. If the individual spin orbitals are... 

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... 

The minimum requirements for a many-electron wave function, namely, antisymmetry with respect to interchange of electrons and indistinguishability of electrons, are satisfied by an antisymmetrized sum of products of one-electron wave functions (orbitals), ( 1),... 

Answer. Orbitals are one-electron wave functions, ). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), 1(1) 2(2) 3(3) NfNe). However, the fact that electrons are fermions also imposes the requirement that the many-electron wave function be antisymmetric toward the exchange of any two electrons. All of the physical requirements, including the indistinguishability of electrons, are met by a determinantal wave function, that is, an antisymmetrized sum of Hartree products, ( 1,2,3,..., Ne) = 1(1) 2(2) 3(3) ( ). If (1,2,3,...,Ne) is taken as an approximation of (1,2,..., Ne), i.e., the Hartree-Fock approximation, and the orbitals varied so as to minimize the energy expectation value,... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

We say that the wave function is antisymmetric. As it turns out, nature demands many-electron wave functions to be antisymmetric. Such an antisymmetric wave function can be written as a determinant ... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

The Hartree-Fock method is a procedure for finding the best "many-electron" wave function If7 as an antisymmetrized product of one-electron orbitals m p)... 

From a purist theoretical point of view, there is one further important result hidden in the Levy constrained-search strategy it provides a unique, albeit only formal, route to extract the ground state wave function F, from the ground state density p0. This is anything but a trivial problem, since there are many antisymmetric N-electron wave functions that yield... 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

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-... 

The many-electron wave function of a molecular system is taken as the antisymmetrized product of (pt, and for closed-shell systems it is convenient to represent it by a Slater determinant. Such an approach is known as the restricted Hartree-Fock (RHF) method and is the most widely used method in chemisorption calculations. Its principal drawback is the neglect of Coulomb electron correlation, which is of crucial importance for adequate treatment of chemical rearrangements with varying numbers of electron pairs. 

A method in which a many-electron wave function is written as an antisymmetric product of one-electron orbitals. The instantaneous electron-electron repulsion is replaced by interaction with the time-averaged densities of the other electrons... 

As the standard ansatz few the many-electron wave function is an antisymmetrized product of one-electron spinors o) it may be written as a Slater determinant or in the language of second quantization (see, for example, Helgaker et al. 2000) as... 

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