Symmetry-adapted wavefunctions

After having identified the symmetry properties of basis sets of atomic orbitals, we need to explore howto build wavefunctions with given symmetry properties from a particular basis. First we need to introduce the concept of symmetry-equivalent atoms or orbitals. Clearly, the two hydrogen Is orbitals of water 4.4 are equivalent, specifically because any symmetry operation of the C2V point group will either send Xi to itself or to xa and likewise send xa to itself or to xi- Thus, xi, Xa form a completely equivalent set The SF4 molecule of 4.7 also belongs to the point group. Here, however, no symmetry operation sends xi or xi to xs or X4. or vice 

We now generate the orbitais of 0 and p2 symmetry for the hydrogen Is orbitals of water. Since the symmetry properties of the resuiting function must satisfy each entry y,( ) of the reievant row of the character tabie, starting with the full complement of symmetry-equivalent orbitals (xi and xa in this case), we force the resuiting wavefunction to have particular symmetry properties by construction using the character tabie itseif. We write 

is any member of a complete symmetry-equivalent set. It does not matter whether we start off with x or Xi in this equation. Since they are symmetry equivalent, the same result will be found. For the 2 point group, the results of the operations /5(xi) nd (xz) are given by 

This is shown pictoriaiiy in 4.8 and may be normalized if desired. For / = b2 then. 

This is shown pictoriaiiy in 4.9. This technique using equation 4.37 is a very general and powerful way to construct functions of this type. The operator in this equation is 

---

Kutzelnigg W 1992 Does the polarization approximation converge for large-rto a primitive or a symmetry-adapted wavefunction Chem. Phys. Lett. 195 77... 

We may disregard the closed-shell cores of the atoms since these play no role in the construction of symmetry-adapted wavefunctions, and concentrate attention upon the valence electrons. In the simplest case, with one valence electron per atom, we have a configuration 0102 n of N singly-occupied, non-degenerate valence orbitals which is then said to form a covalent structure for the molecule. Then under any spatial symmetry operation (%, a VB function Vsu-.k transforms as... 

The small reps of little groups are sufficient for many purposes in sohd-state theory, such as classifying states within electron energy bands and vibration frequencies within phonon band spectra, and for generating the symmetry-adapted wavefunctions. 

A.C. Hurley, R.D. Harcourt and P.R. Taylor, Israel J. Chem 19, 215 (1980) use group theoretical projection operators to generate symmetry-adapted wavefunctions for O2. 

It has always been our position to use no more than moderately size d wavefunction expansions. Current limits are about 20,000 symmetry adapted wavefunctions built from fewer than 1 million Slater determinants, and use of two virtuals per /, per shell (n). This allows the physics (systematics) to be more visible and reduces the need for large computational resources that were frequently unavailable in the old days. Development of systematic rules is one of the main goals of our research. Some examples follow (i) determining which correlation effects are most important for a specitic property [14,15], (ii) near conservation of f-value sums for nearly degenerate states [15,16], (iii) similar conservation of g-value sums [16], (iv) similar conservation of magnetic dipole hyperfme constants [17,18]. This approach does mean near maximal use of symmetry, creating extra auxiliary... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

The GUGA-Cl wavefunctions are spatial and spin symmetry-adapted, thus the projections of total orbital angular momentum and total spin of a hydrogen molecule in a particular electronic state are conserved for all the values of R. Therefore, the term remains constant for an electronic state, and it causes a... 

Symmetry adaptation ofVB wavefunctions Defining the (idempotent) projection operator... 

Because symmetry operators commute with the electronic Hamiltonian, the wavefunctions that are eigenstates of H can be labeled by the symmetry of the point group of the molecule (i.e., those operators that leave H invariant). It is for this reason that one constructs symmetry-adapted atomic basis orbitals to use in forming molecular orbitals. 

In summary, proper spin eigenfunctions must be constructed from antisymmetric (i.e., determinental) wavefunctions as demonstrated above because the total S2 and total Sz remain valid symmetry operators for many-electron systems. Doing so results in the spin-adapted wavefunctions being expressed as combinations of determinants with coefficients determined via spin angular momentum techniques as demonstrated above. In... 

One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy V is unchanged when ah of the electrons have their position vectors subjected to inversion (i r = -r). This quantum number is straightforward to determine. Because each L, S, Ml, Ms, H state discussed above consist of a few (or, in the case of configuration interaction several) symmetry adapted combinations of Slater determinant functions, the effect of the inversion operator on such a wavefunction P can be determined by ... 

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2V symmetry. The configuration which dominates the ground-state wavefunction has doubly occupied O and C Is orbitals, two CH bonds, a CO o bond, a CO 7t bond, and two O-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (lai22aj23ai2lb22... 

In the MO formalism it is quite straightforward to deal with the excited states of a molecule. An adequate wavefunction of an excited state can be constructed according to the resultant configuration and its symmetry arising from electron promotion among MO series. Compared with numerous MO-based methods, VB approaches are far less employed to study excited states due to the difficulty in VB computations. Recently, by observing the correlation between MO theory and resonance theory, as well as the symmetry-adapted VB wavefunction described in the last section, we performed VB calculations on low-lying states of some molecules [71, 72],... 

The final wavefunction for BH3 can thus be said to consist of two fully-symmetric configurations, one of them essentially a double excitation out of the other, and a symmetry-adapted linear combination of six equivalent low-symmetry configurations. The latter can be viewed as distortions of the main configuration. The wavefunction therefore includes only three truly independent configurations, and is thus readily amenable to physical interpretation, while achieving an accuracy that vouches for the significance of such interpretations. 

The introduction of the concept of one-electron crystal orbitals (CO s) considerably reduces difficulties associated with the many-electron nature of the crystal electronic structure problem. The Hartree-Fock (HF) solution represents the best possible description of a many-electron system with a one-determinantal wavefunction built from symmetry-adapted one-electron CO s (Bloch functions). The HF approach is, of course, only a first approximation to the many-particle problem, but it has many advantages both from practical and theoretical points of view ... 

Let trial wavefunction expressed in terms of r and R relative to R. The preceding considerations lead to the conclusion that the symmetry-adapted 0 (r,R) must be of the form... 

---