Wave function product

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

If the given Hamiltonian operates on the product wave function, ft only operates on ip(i), eic., and the contribution of this wave function to the expectation value of the energy becomes... 

However, if we restrict the form of 4> by model approximations then we can no longer guarantee that the variations 6 R will be such as to maintain the symmetry of the total product wave function... 

This expression is just the one which obtains for the Hartree product wave-function. The difference between this Hartree wavefunction and the Fock wavefunction of Eq. (1) is the absence of the antisymmetrizer j4 in that equation. This means that in the Hartree wavefunction each electron can be identified with a specific molecular orbital, whereas in the Fock wavefunction all electrons make use of all orbitals. The Hartree wavefunction is of course not a proper quantum mechanical wavefunction, since it is not antisymmetric in the electrons. Moreover, for the Fock wavefunction, it is in general not possible to reduce the interorbital exchange energy to zero. But the localized molecular orbitals, as defined here, represent that set of molecular orbitals for which the energy expression comes closest to the Hartree form, i.e. they come closest to being identifiable with electrons which are not exchanged among different orbitals. 

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

A wave function of die form of Eq. (4.35) is called a Hartree-product wave function. The eigenvalue of T is readily found from proving the validity of Eq. (4.35), viz.. 

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian That is, we wish to find orbitals that minimize (4 hp H I hp). By applying variational calculus, one can show that each such orbital i/f, is an eigenfunction of its own operator hi defined by... 

At this point it is appropriate to think about our Hartree-product wave function in more detail. Let us say we have a system of eight electrons. How shall we go about placing them into MOs In the Hiickel example above, we placed them in the lowest energy MOs first, because we wanted ground electronic states, but we also limited ourselves to two electrons per orbital. Why The answer to that question requires us to introduce something we have ignored up to this point, namely spin. 

Knowing these aspects of quantum mechanics, if we were to construct a ground-state Hartree-product wave function for a system having two electrons of the same spin, say a, we would write... 

Slater determinants have a number of interesting properties. First, note that every electron appears in every spin orbital somewhere in the expansion. This is a manifestation of the indistinguishability of quantum particles (which is violated in the Hartree-product wave functions). A more subtle feature is so-called quantum mechanical exchange. Consider the energy of interelectronic repulsion for the wave function of Eq. (4.43). We evaluate this as... 

H — H°. The perturbation H contains interaction terms between the electronic, vibrational, and rotational motions, and adds a correction term to the simple product wave function ... 

With a similar wave-function for He atom B the product wave-function will be... 

Another rationalization of the exponential form of the wave-function is obtained by considering two non-interacting N-electron systems. If a wave-function of type (l+Ti o is tried, the product wave-function is... 

Product wave functions can clearly be constructed for any number of electrons. Early wave functions, were constructed on this basis together with the empirical rule that not more than two electrons could be assigned to a single orbital, one of each. spin. Further, electrons tend to occupy the orbitals with, lowest possible energy in the absence of other factors. 

Another well-known property of determinants is that they vanish if they have two identical rows. This means that it is not possible to construct a non-vanishing antisymmetrised product in which two electrons in the same orbital have the same spin. Thus the rule that not more than two electrons must be assigned to any one space orbital follows as a direct consequence of the antisymmetry principle for product wave functions it had to be introduced as an extra postulate. 

The wavefunction T of the autoionizing state is given by Eq. (21.1), and to calculate the photoionization cross section we need the dipole matrix element ( bl l ) = (Tb /i A202)- We can write (Tb and 02) as product wave functions ... 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

In molecular crystals the forces between molecules are much smaller than those within molecules. It is for this reason adequate in dealing with the low electronically excited states to use the free-molecule wave functions and to treat crystal perturbations on them only in a higher approximation. Accordingly we shall denote by q>, y1, q ,.. ., electronic states of the free molecules, and show by subscripts (i,j) the location of the chosen molecule in the crystal. In a crystal of N molecules with h in each unit cell, the complete set of molecules is contained in the product wave function... 

Exercise 6.5 Let us consider the general process X + H—Y —> X—H + Y, with X = Y. At the transition state geometry, the ground state is the normalized combination of the reactants and products wave functions, d>i and d>2- T° facilitate the derivation, let us write these two wave functions so that their overlap is positive, that is,... 

A configurational wave function (here denotes the molecular electronic eigenfunction xFe) is represented by an antisymmetrized product wave function [5] ... 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

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