Eigenfunction spatial

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

The spin in quantum mechanics was introduced because experiments indicated that individual particles are not completely identified in terms of their three spatial coordinates [87]. Here we encounter, to some extent, a similar situation A system of items (i.e., distributions of electrons) in a given point in configuration space is usually described in terms of its set of eigenfunctions. This description is incomplete because the existence of conical intersections causes the electronic manifold to be multivalued. For example, in case of two (isolated) conical intersections we may encounter at a given point m configuration space four different sets of eigenfunctions (see Section Vni). 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

The Time Reversal Operator.—In this section we show that spatial operators are linear whereas the time reversal operator is antilinear.5 This may be seen by examining the eigenfunctions of the time dependent Schrodinger equation... 

The value o+l <0.4 found for H2 shows that even in the lowest state the molecules are rotating freely, the intermolecular forces producing only small perturbations from uniform rotation. Indeed, the estimated (3vq<135° corresponds to Fo <28 k, which is small compared with the energy difference 164 k of the rotational states j = 0 and j= 1, giving the frequency with which the molecule in either state reverses its orientation. The perturbation treatment shows that with this value of Fo the eigenfunctions and energy levels in all states closely approximate those for the free spatial rotator.9... 

Thus, the spatial function (q) is actually a set of eigenfunctions t/ n(q) of the Hamiltonian operator H with eigenvalues E . The time-independent Schrodin-ger equation takes the form... 

The basis of the expansion, ifra, are configuration state functions (CSF), which are linear combinations of Slater determinants that are eigenfunctions of the spin operator and have the correct spatial symmetry and total spin of the electronic state under investigation [42],... 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

Solution (15) is written in the form of a normal mode expansion with eigenvalues —X D and eigenfunctions represented by the spatial parts of Eq. (15). Of course, the above solution must fulfil the appropriate initial and boundary conditions. 

The overlap of ip with the true ground state eigenfunction ipo is greater than or equal to 1 — e that is, the spatial distribution of the trial wave function is a very good approximation to the true wave function, and... 

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

UHF, on the other hand, does optimize tire a and orbitals so tlrat they need not be spatially identical, and thus is able to account for both spin polarization and some small amount of configurational mixing. As a result, however, UHF wave functions are generally not eigenfunctions of the operator S-, but are contaminated by higher spin states. 

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