Spatial function symmetry

Spatial function Spin function Symmetry Configuration... 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Figure 1.1 also shows/i,/2, and/3. All of these functions rapidly approach zero for large values of x. Functions like this are entirely appropriate for representing the wave function or electron density of an isolated atom. This example incorporates the idea that we can combine multiple individual functions with different spatial extents, symmetries, and so on to define an overall function. We could include more information in this final function by including more individual functions within its definition. Also, we could build up functions that describe multiple atoms simply by using an appropriate set of localized functions for each individual atom. 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

In the present case we have ten AO basis functions, and these provide a set of 55 symmetric (singlet) spatial functions. Only 27 of these, however, can enter into functions satisfying the spatial symmetry, of the ground state of the H2 molecule. Indeed, there are only 14 independent linear combinations for this subspace from the total, and, working in this subspace, the linear variation matrices are only 14 x 14. We show the energy for this basis as the lowest energy curve in Fig. 2.8. We will discuss the other curves in this figure later. 

Now we can consider the symmetry properties of the spatial functions corresponding to the above spin functions. They are uniquely defined from the requirement that the product of the spatial and spin functions must be antisymmetric. In a way, what was symmetrized in the spin part (rows) must be antisymmetrized in the spatial part (columns) and vice versa. That means that the spin function represented by a two-row Young pattern T with the first row longer by 2S boxes than the second one must be complemented by a spatial function represented by a two-column Young pattern T with the first column longer by 2S boxes than the second one, e.g. ... 

Thus beginning with an arbitrary spatial function O, we form from it a set of functions with the correct spatial symmetry according to (14), and finally the full antisymmetric function (7). This is all incorporated in expression (12) on which we now focus attention. 

Consequently two normalised, symmetry-adapted spatial functions result... 

The E4T option to the UQCISD method keyword tells Gaussian to run fhe component MP4 calcdations at the MP4(SDTQ) level, rather than the default of MP4(SDQ). When you set up the QCISD(T) calculations for methane, include the additional option IOP(2/16=1) in the route section (which says to ignore any symmetry changes during the scan) and also include Gue s=(Always b() for the unrestricted casr. Mix requests that the HOMO and LUMO be mixed so as to destroy alpha-beta and spatial spin symmetries (this option is also useful for producing unrestricted wave functions for singlet systems), and Always says to recompute a new guess wavefunction at each point. 

Notice that, since the basis functions are spatial functions the basis-function electron-repulsion integrals involve no spin integration if one of them is zero it is because of symmetry or simply a numerical accident because the two charge-clouds are very remote. 

The exact spatial-spin eigenfunctions of a system should belong to representations of the direct product group of spatial and spin symmetry operators. The Hamiltonian including spin is invariant under all the spatial-spin symmetry transformations. Thus the spin-orbit operator (unspecified) is invariant under all space-spin symmetry transformations. The total symmetry of the eigenfunctions is the direct product of the symmetries of the spatial and spin parts of the function. Thus, if we can obtain the irreducible representations of the spin functions, the spin-orbit selection rules will be known from... 

Besides functions, we must also consider the action of operations on operators. In quantum chemistry, operators, such as the Hamiltonian, H, are usually spatial functions and, as such, are transformed in the same way as ordinary functions, e.g.. So why devote a special section to this Well, operators are different from functions in the sense that they also operate on a subsequent argument, which is itself usually a function. Hence, when symmetry is applied to an operator, it will also affect whatever follows the operator. Symmetry operations act on the entire expression at once. This can be stated for a general operator O as follows ... 

This theorem reflects the influence of time-reversal symmetry. Already in Chap. 2, we showed that the time-reversal operator is an anti-linear operator. It will turn any spatial wavefunction into its complex conjugate. Applying it twice in succession will return the original wavefunction, and, hence, we may write for spatial functions ... 

The first two functions in (13.89) are symmetric. They therefore go with the antisymmetric singlet spin function (11.60). Clearly, these two spatial functions have different energies. The last two functions in (13.89) are antisymmetric and hence are the spatial factors in the wave functions of the two 2 terms. The four functions in (13.89) are found to have eigenvalue +1 or -1 with respect to reflection of electronic coordinates in the xz cr symmetry plane containing the molecular (z) axis (Prob. 13.30). The superscripts + and - refer to this eigenvalue. 

In evaluating the elements of H and M, the special form of the spatial function may now be recognized. By assuming that electrons 1 and 2 occupy the first orbital, 3 and 4 the second, and so on, we impose a symmetry on the spatial function 0. If 0 is symmetric under transposition (12), it will be necessary to ensure that the spin factor is anrisymmetric under (12) this must be so for each doubly occupied orbital, and the first g columns of any Young tableau describing an associated spin eigenfunction will thus read... 

The above equivalence confirms that, with a spinless Hamiltonian, we may obtain the same energy expectation value either (i) by expanding (7.4.1) in terms of determinants and then using the rules in Section 3.3 or (ii) by using a linear combination of purely spatial functions (7.4.6) of appropriate symmetry. The second approach is essentially that of spin-free quantum chemistry (Matsen, 1964), which is considered in more detail in later sections. In the present case a first-principles argument will lead to the required matrix-element expressions. 

KrOnecker product U(m) x SU(2). We have seen, however, that the orbital and spin factors in a wavefunction can always be separated provided we insist that the spatial functions have the appropriate symmetry (under electron permutations), so that they may be combined with spin eigenfunctions (as in Section 4.3) to yield space-spin wavefunc-tions that satisfy the Pauli principle. In the present chapter we shall not need the spin factors we consider the space spanned by the orbital products (10.2.13), and as in Section 7.6 seek linear combinations, from all possible configurations, that possess appropriate symmetry under electron permutations. 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

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. 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

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