Function symmetry adapted

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04 , where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 II 0 ) = ( b H 04 ), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or 

The representation of the A-electron symmetry group generated by a given configuration is block-diagonalized into irreducible representations indexed by A, with subspecies index M. In general there will be some number m n of independent functions with the same symmetry indices. If the configuration basis is ,, i = 1, / , then a set of m n unnormalized projected determinants = 0 f j = l 5Z =1 are to be constructed, where O is a projector onto the symmetry species A, M. The function j j is normalized if j = YH=i yp I2 and yjj = 1. The projected determinants can be transformed into an orthonormal set 4V = k fi Ya=i xtu where kn = E = i xm I2- Alternatively, as an expansion in 

In the basis of these orthonormal symmetry-adapted functions, matrix elements of the invariant Hamiltonian are given for two different configurations A and B by 

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The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

Prove that if the irreducible representation r, is one dimensional, then the function g of (9.67) is a symmetry-adapted function that transforms according to r,. Start by writing the basis function / as some linear combination of symmetry-adapted functions g, apply Or to this equation then multiply by x (R) and sum over R. 

Because the group is of infinite order, (9.67) is cumbersome to use, and it is simplest to find the symmetry-adapted functions by inspection. Each symmetry operation either leaves lja and sb alone or transforms them into each other, and the following functions are easily seen to transform... 

These operators are examples of projection operators they project out the symmetry-adapted functions from the basis AOs.) Application of PAt to l o gives... 

Note, however, that since we now work with only the trace of the matrix, we have no information about off-diagonal elements of the irrep matrices and hence no way to construct shift operators. The business of establishing symmetry-adapted functions therefore involves somewhat more triad and error than the approach detailed above. Character projection necessarily yields a function that transforms according to the desired irrep (or zero, of course), but application of character projection to different functions will be required to obtain a set of basis functions for a degenerate irrep, and the resulting basis functions need not be symmetry adapted for the full symmetry species (irrep and row) obtained above. 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

As was discussed in Chapter 2, the need to have full matrix representations available to obtain basis functions adapted to symmetry species is something of a handicap. Although character projection itself is not adequate for this task, Hurley has shown how the use of a sequence of character projectors for a chain of subgroups of the full point group can generate fully symmetry-adapted functions. Further discussion of this approach is beyond the scope of the present course, but interested readers may care to refer to the originad literature [6]. 

Note that the average over k on the RHS of Eq. 4.5 means that the integral itself is independent of the value of the row index i. Thus the matrix of O in a basis of symmetry-adapted functions is diagonal on symmetry species, and the diagonal blocks corresponding to rows of the same irrep are the same. 

Symmetry-adapted functions Table 4.1. a/x table for functions 2S (abc)... 

Nesbet, R.K. (1961). Construction of symmetry-adapted functions in the many-body problem, J. Math. Phys. 2, 701-709. 

The symmetry-adapted functions themselves may then be constructed directly from equations (59), (60), and (61). The same procedure can be applied without any difficulty to the other types of ionic configuration. The results show that there are altogether a total of 268 multiplets of which 22 correspond to xAig states. A similar result, of course, would be obtained from MO theory with full configuration interaction.33... 

We have named the elements of the matrix D <91) group angular overlap integrals (20). For the particular case of symmetry adapted functions, as here, the elements of 2>aI> <31) are the so-called group overlap coefficients, or the ratio between the group overlap integral and the diatomic overlap integral. 

The different energy values are fitted to a symmetry adapted functional form as described before by a least square procedure when geometry is optimized at each conformation, the rotation angles lose their simple meaning, but the dynamical symmetry of the molecule remains, even when this molecule appears to be deformed during the optimization process. 

When, the geometry of the molecule is fully optimized at each conformation, the geometry of the whole molecule changes during the rotation. As a result, the rotational constants change with the rotation angles, and have to be fitted in a Fourier series. Since the deformation does not influence the dynamical symmetry properties of the molecule, the rotational B constants may be fitted to a symmetry adapted functional form identical to that of the potential energy function of (113). 

In calculations based on the MO-LCAO technique [32-34], the one-electron Kohn-Sham equations Eq. (11) are solved by expanding the molecular orbital wavefunctions V i(r) in a set of symmetry adapted functions Xj(r), which are expanded as a linear combination of atomic orbitals i.e. 

Separability theorem, 309 SHAKE algorithm, 385 SHAPES force field, 40 Simulated Annealing (SA), global optimization, 342 Simulation methods, 373 Supidfiidiil, iulcs, 3j6 Susceptibility, 237 Symbolic variables, for optimizations, 416 Symmetrical orthogonalization of basis sets, 314 Symmetry adapted functions, 75 Symmetry breaking, of wave functions, 76 ... 

By forming suitable linear combinations of basis functions (symmetry adapted functions), many one- and two-electron integrals need not be calculated as they are known lu be exactly zero due to symmetry. Furthermore, the Fock (in an HP calculation)... 

B. Intermolecular Potential in Terms of Symmetry-Adapted Functions. 137... 

C. Expansion of Atom-Atom Potentials in Symmetry-Adapted Functions. 141... 

The second symmetry requirement that the expression for the inter-molecular potential has to meet is that it must be invariant under any rotation of the global coordinate frame. The transformation properties of the symmetry-adapted functions Gj Hw) under such a rotation are easily obtained from Eqs. (10) and (5) ... 

The summation labels are defined as 1 = /i, l2, h and m = mi, m2, m3. We note again that for certain values of / and l2 more than one symmetry-adapted function G exists and that in this case 2, includes all these functions. 

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