Invariance with respect to a unitary transformation

MP2-R12 with a single coefficient c is invariant with respect to a unitary transformation of the pair functions. 

Having pointed out that a single Slater determinant is invariant with respect to a unitary transformation among the occupied orbitals, we can come to the idea of bent (or t) bonds (see e.g. 38>). The double bond in ethylene is normally described by a a and a 71 bond, both of them localized between the C atoms. Now, a completely equivalent description can be used where the bonding a and n orbitals are replaced by their normalized sum and difference... 

The optimization method outlined in Sections II.C-E shows very fast and stable convergence behaviour when applied to CASSCF wavefunctions, in which all orbital rotations between occupied orbitals are redundant. However, experience has shown that convergence is often much slower when orbital rotations between strongly occupied valence orbitals have to be optimized. It has been shown that this is due to the fact that the energy approximation (T) is not invariant with respect to a unitary transformation between two doubly occupied orbitals . If the 2 x 2 transformation in Eq. (34) is applied to a single Slater determinant with just two doubly occupied orbitals, the second-... 

Whereas the lEPA-scheme means physically that one treats one electron pair in the Hartree-Fock field of the other electrons, CEPA means that each pair is treated in the fields of the correlated other electrons which is somewhat more physical. Through the explicit consideration of the Bab block the interaction between the different pair correlations is accounted for. Though a rigorous justification of the CEPA-method is still lacking and probably not possible the practical applications have so far been very satisfactory. One main advantage of CEPA as compared with lEPA is that it can also be used with delocalized orbitals and the results are nearly invariant with respect to a unitary transformation of the occupied orbitals. 

These terms are nowhere included within the lEPA, which is the major drawback of this procedure. As a consequence, one finds that evaluated according to Eqs. (5) and (6) depends crucially on the actual choice of occupied molecular orbitals (MOs), i.e. is not invariant with respect to a unitary transformation of occupied MOs. lEPA correlation energies may either overshoot or undershoot the exact E dejiending on the chosen. This deficiency may be rectified by an inclusion of (9) in in the lowest order of perturbation theory,but this approach will not been pursued further. 

This reasoning can be supported in another way. The CP-MET equations are invariant with respect to a unitary transformation among the occupied and/or the virtual MOs, and one can work in a localized description. The... 

Coulombic operator (p. 337) exchange operator (p. 337) invariance with respect to a unitary transformation (p. 340)... 

It should be noted that, because of the invariance properties of the density function with respect to a unitary transformation among its orbitals, the a-n description of double or triple bonds in terms of nonequivalent orbitals is not the only possible one a description in terms of two or three equivalent bent banana bonds is possible as well. It is... 

The operators 7 and K are invariant with respect to any unitary transformation of the spinorbitals. In conclusion, while deriving the new spinorbitals from a unitary transformation of the old ones, we do not need to worry about 7 and K since they remain the same. 

The basic concept of this method was given by Lennard-Jones and Pople and applied by Edmiston and Ruedenberg. ° It may be easily shown that for a given geometry of the molecule, the functional Jij is invariant with respect to any unitary transformation of the orbitals ... 

It turned out that this ansatz is not invariant with respect to a switch from canonical to localized orbitals, and that localized orbitals are actually the better choice [13]. This lack of unitary invariance is, of course, unsatisfactory, and one of us [14] has proposed a generalization in which orbital invariance is established. We come to this later. Let us first apply a pair transformation to (96). 

It is important to mention here that for any practical implementation of a UGA scheme, one need not demand that the n-electron CSF s be adapted to U n). Rather, it is expedient to demand that the CSF s be adapted to the subgroup of U n),u n) = U ric) <8> U tia) U nv), where Tic, na, and v are. respectively, the number of core, active, and virtual orbitals. This is simply due to the physical requirement that the maximum invariance of an approximate function that one may practically impose is the invariance of the function and the energy with respect to separate unitary transformation among core, active, and virtual orbitals. This was indeed done in the UGA CC papers by Paldus and others [33, 34], In particular, both these papers have used the h( )-adapted scalar tensor generators in their choice of excitation operators. We also point out two references [35, 36] in this context where useM discussions of other UGA-based approaches and their interrelation can be found. Reference [36] has also presented in considerable detail a number of approximately spin-adapted CC approaches and their relationship with UGA CC. We also point out that Li and Paldus have applied the UGA CC method to many problems (see e.g.. Refs. 39 8 in [37]). 

However, as is well known, the trace is invariant with respect to choice of basis functions that are related by a unitary transformation. Thus, rather than working with the basis of eigenfunctions we may, following eqn (4.31), work with respect to the basis of atomic orbitals, to write... 

This situation is of course not satisfactory as observable quantities should be invariant with respect to unitary basis transformations. " Here, we outline the adiabatic route to a basis-invariant formulation of the theory. 

It would be good now to get rid of the non-diagonal Lagrange multipliers in order to obtain a beautiful one-electron equation analogous to the Fock equation. To this end, we need the operator in the curly brackets in Eq. (11.33) to be invariant with respect to an arbitrary unitary transformation of the spinorbitals. The sum of the Coulomb operators (Ucoui) is invariant, as has been demonstrated on p. 406. As to the unknown functional derivative SE c/Sp (i.e., potential Uxc), its invariance follows from the fact that it is a functional of p [and p of Eq. (11.6) is invariant]. Finally, after applying such a unitary transformation that diagonalizes the matrix of Sij,v/e obtain the Kohn-Sham equation (su = Si) ... 

Among the usual advantages of such expressions as Eq. (7-80) and (7-81), one is salient they show forth the invariance of p and w with respect to the choice of the basis functions, u, in terms of which p, a, and P are expressed. The trace, as will be recalled, is invariant against unitary transformations, and the passage from one basis to another is performed by such transformations. The trace is also indifferent to an exchange of the two matrix factors, which is convenient in calculations. Finally, the statistical matrix lends itself to a certain generalization of states from pure cases to mixtures, required in quantum statistics and the theory of measurements we turn to this question in Section 7.9. 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

Here inv stands for an invariant in respect to transformation consistent with the symmetry of the system. For quantum mechanical operators, this means unitary transformations. The parameter Ae in Eq. [107] quantifies the extent of mixing between two adiabatic gas-phase states induced by the interaction with the solvent. For a dipolar solute, it is determined through the adiabatic differential and the transition dipole moments... 

The invariance of the first-order density matrix with respect to unitary transformations ensures the invariance of all one-electron properties, like electrostatic potentials. Thus the transformation to localized orbitals does not alter the value of the potential at any point r of the space, but permits a chemically meaningful partition of this quantity. In fact, the lone pair , bond and core localized orbitals resulting from the Boys transformation are particularly suitable for our attempt a) to give a rational basis to the additivity rules for group contributions, and b) to find some criteria by which to measure the degree of conservation of group properties. 

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