Function, antisymmetric

It is essential to realize that the energies (i H Oi> of the CSFs do n represent the energies of the true electronic states Ek the CSFs are simply spin- and spatial-symmetry adapted antisymmetric functions that form a basis in terms of which to expand the true electronic states. For R-values at which the CSF energies are separated widely, the true Ek are rather well approximated by individual (i H Oi> values such is the case near Rg. 

The diffusion and Greens function Monte Carlo methods use numerical wave functions. In this case, care must be taken to ensure that the wave function has the nodal properties of an antisymmetric function. Often, nodal sur-... 

The simplest antisymmetric function that is a combination of molecular orbitals is a determinant. Before forming it, however, we need to account for a factor we ve neglected so far electron spin. Electrons can have spin up i+Vi) or down (-V2). Equation 20 assumes that each molecular orbital holds only one electron. However, most calculations are closed shell calculations, using doubly occupied orbitals, holding two electrons of opposite spin. For the moment, we will limit our discussion to this case. 

In other words, the energy of the exact wavefunction serves as a lower bound to the energies calculated by any other normalized antisymmetric function. Thus, the problem becomes one of finding the set of coefficients that minimize the energy of the resultant wavefunction. 

Boys, S. F., Proc. Roy. Soc. London) A206, 489, Electronic wave functions. III. Some theorems on integrals of antisymmetric functions of equivalent form/ ... 

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative... 

In Equation 1.3, the radial function Rnl (r) is defined by the quantum numbers n and l and the spherical harmonics YJ" depend on the quantum numbers l and W . When the spin of the electron is taken into account, the normalized antisymmetric function is written as a Slater determinant. The corresponding eigenvalues depend only on n and l of each single electron, which determine the electronic configuration of the system. 

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

Because subsystems A and B do not interact, it must be that T a consists of a determinantal expansion in functions taken solely from the set Ha, and similarly uses only those spin orbitals in Br. It follows that T a and are strongly orthogonal [53]. Two antisymmetric functions f x, ..., Xp) and g yi,..., yg) are said to be strongly orthogonal if... 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

As an example, consider H2. The nuclear spin of H is and we have three symmetric nuclear spin functions and one antisymmetric function. The symmetric spin functions are of the form (1.251)—(1.253), and correspond to the two nuclear spins being parallel. Designating the quantum number of the vector sum of the two nuclear spins as 7, we have 7= 1 for the symmetric spin functions. The antisymmetric spin function has the form (1.254), and corresponds to 7 0. The ground electronic state of H2 is a 2 state, and the nuclei are fermions hence the symmetric (7=1) nuclear spin functions go with the J= 1,3,5,... rotational levels, whereas the 7=0 spin function goes with the7=0,2,4,... levels. 

Functions having the property f -x) =,f x) are called symmetric, or even, functions, whilst those having the property f(-x) — -f(x) are called antisymmetric or odd functions. In our discussion of trigonometric and hyperbolic functions, we have encountered a number of examples of functions that fall into one or other of these categories, as well as some that fall into neither. Symmetric and antisymmetric functions are so called because they are symmetric or antisymmetric with respect to reflection in the y-axis. A close look at Figure 2.17 shows that, since cos0=cos(-0), and sin0=-sin(-0), the cos and sin functions are symmetric and antisymmetric, respectively. Likewise, we can classify the cosh and sinh functions... 

Antisymmetrized function (10.8) has the property that if any two one-electron functions are identical, then xp is identically zero (satisfying the Pauli exclusion principle). Its second very important property if any two electrons lie at the same position, e.g., ri = r2 (and they also have parallel spins Si = S2), then P = 0. As the functions

spatial variables (r,9,with parallel spin are close together. Thus, unlike the single product function, the antisymmetrized sum of product functions (10.8) shows a certain degree of electron correlation. This correlation is incomplete - it arises by virtue of the Pauli exclusion principle rather than as a result of electrostatic repulsion, and there is no correlation at all between two electrons with antiparallel spins [16]. 

The antisymmetric wave function for N electrons (0 < N < 2(2j + 1)) in the isospin basis can be constructed out of conventional antisymmetric functions obtainable by vectorial coupling of momenta of individual shells, i.e. 

This method may also be applied to the calculation of polarizabilities. Indeed, we arrived at the present formulation of this method when wishing to perform such calculations. More recently,7 it has been applied to the problem of calculating the polarizability of ions having the rare gas structure. Here again the ground state is considered as being described by an antisymmetric function composed of Slater orbitals. The excited orbital was taken to be of the form ... 

Moreover, since the antisymmetric functions have the forms (P integral)... 

The //-electron wave function T is an antisymmetric function of N sets of spatial and spin coordinates r,, v, for individual electrons, all evaluated at a common timer. In postulating a time-dependent Schrodinger equation of the form... 

Also, such an antisymmetric function automatically satisfies the Pauli Exclusion Principle. For instance, if we write a function for helium having both electrons in the Is orbital with spin up,... 

The total, antisymmetric function for a closed-shell configuration is expressed as a Slater determinant built-up from the spin-orbitals. In the case of open-shell configurations, a linear combination of Slater determinants may be needed in order to obtain a function with the same symmetry and multiplicity characteristics as the state under consideration. 

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