Integral symmetry

CO, CO, co, and o, respectively. The integrals in Eqs. (E.9) and (E.IO) will then be different from zero only if the integrands are invariant under all symmetry operations allowed by the symmetry point group, in particular under C3. It is readily seen that the linear terms in Q+ and Q- vanish in and H In turn. 

After considering the symmetry of the two-electron integrals, we have... 

In practice, in order to maintain the symmetry of elemental coefficient matrices, some of the first order derivatives in the discretized equations may also be integrated by parts. 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Assuming symmetry of vj and V2 with respect to mid-surface at = 0), the velocity components, given by Equation (5.70), are integrated to obtain flow rates in the lateral directions within the limits of the thin layer as... 

If we expand Eq. (10-7) and simplify aeeording to the symmetry of the problem, (Richards and Cooper, 1983) the integral breaks up in the way it did for the helium atom excited state... 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Here, the indices i and k, which label the spin-orbital having variables r are grouped together, and j and 1, which label spin-orbitals referring to the r variables appear together. The above permutational symmetries, when expressed in terms of the Mulliken integral list read ... 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

This relationship, when used, for example, in the symmetrie or spherieal top El integral ... 

The first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

The second term in the above expansion of the transition dipole matrix element Za 3 if i/3Ra (Ra - Ra,e) can become important to analyze when the first term ifi(Re) vanishes (e.g., for reasons of symmetry). This dipole derivative term, when substituted into the integral over vibrational coordinates gives... 

An example of an El forbidden but "vibronically allowed" transition is provided by the singlet n ==> ti transition of H2CO that was discussed earlier in this section. As detailed there, the ground electronic state has Ai symmetry, and the n ==> 71 state is of 1A2 symmetry, so the El transition integral... 

These density matriees are themselves quadratie funetions of the CI eoeffieients and they refleet all of the permutational symmetry of the determinental funetions used in eonstrueting F they are a eompaet representation of all of the Slater-Condon rules as applied to the partieular CSFs whieh appear in F. They eontain all information about the spin-orbital oeeupaney of the CSFs in F. The one- and two- eleetron integrals < I f I > and < (l)i(l)j I g I (l)ic(l)i > eontain all of the information about the magnitudes of the kinetie and Coulombie interaetion energies. 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

The role of symmetry in determining whether such integrals are non-zero can be demonstrated by noting that the integrand, considered as a whole, must contain a component that is invariant under all of the group operations (i.e., belongs to the totally... 

There is no component of ai symmetry in the triple direct product, so the integral... 

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