Unitary transformation, invariance

Such a compact MCSCF wavefunction is designed to provide a good description of the set of strongly occupied spin-orbitals and of the CI amplitudes for CSFs in which only these spin-orbitals appear. It, of course, provides no information about the spin-orbitals that are not used to form the CSFs on which the MCSCF calculation is based. As a result, the MCSCF energy is invariant to a unitary transformation among these virtual orbitals. 

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. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

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... 

As well known, this condition is not a restriction whenever the wavefunction is invariant under an unitary transformation [2],... 

If the Lm and Ln m subspaces contain further invariant (proper) subspaces within them the process of reduction can be carried on until no further unitary transformation can be found to further reduce the matrices of the representation. The final form of the matrices of the representation T may... 

Invariance of the trace of a matrix under unitary transformation corresponds to the invariance of phase density under canonical transformation in classical theory. 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

M. D. Benayoun and A. Y. Lu, Invariance of the cumulant expansion under l-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387, 485 (2004). 

In this manner, we have arrived at the Pernal nonlocal potential [81]. It can be shown, using the invariance of Vee with respect to an arbitrary unitary transformation and its extremal properties [13] or by means of the first-order perturbation theory applied to the eigenequation of the 1-RDM [81], that the off-diagonal elements of Uee may also be derived via the functional derivative... 

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... 

Due to the spin polarization effect, the magnetic orbitals can be difficult to identify from a spin-unrestricted calculation. Since the total energy of a Kohn—Sham determinant is invariant under unitary transformations between the spin-up orbitals among each other and spin-down orbitals among each other, one can arrange each spin-up orbital to overlap at most with each spin-down orbital on the basis of the corresponding orbital transformation (COT) (88—90). Then, the molecular orbitals (MOs) are ordered into pairs of maximum similarity between spin-up and spin-down orbitals and can be separated into three groups (i) the MOs with spatial overlap close to one (doubly occupied MOs),... 

Very similar in spirit to CEPA, but formulated as a functional to be made stationary, is the coupled-pair functional (CPF) approach of Ahlrichs and co-workers [28]. CPF can be viewed as modifying the CISD energy functional to obtain size-extensivity for the special case of noninteracting two-electron systems. One disadvantage of some of the CEPA methods is that, unlike CISD or CCSD, the results are not invariant to a unitary transformation that mixes occupied orbitals with one another. CPF... 

Gdanitz and Ahlrichs devised a simpler variant of CPF, the averaged coupled-pair functional (ACPF) approach [30]. This produces results very similar to CPF for well-behaved closed-shell cases and is completely invariant to a unitary transformation on the occupied MOs. Its big advantage is that it can be cast in a multireference form. Multireference ACPF is probably the most sophisticated treatment of the correlation problem currently available that can be applied fairly widely, although it can encounter difficulties with the selection of reference spaces, as discussed elsewhere. 

We may also note here that the MCPF method initially gave a binding energy of only about 8 kcal/mol in the larger basis. This result is in error because MCPF is not invariant to a unitary transformation on the occupied orbitals. Our calculations were performed originally in C2 , symmetry, and at long distances the two Be 2s-derived orbitals that transform according to the aj irreducible representation can mix... 

SU(n) Group Algebra. Unitary transformations, U( ), leave the modulus squared of a complex wavefunction invariant. The elements of a U( ) group are represented by n x n unitary matrices with a determinant equal to 1. Special unitary matrices are elements of unitary matrices that leave the determinant equal to +1. There are n2 — 1 independent parameters. SU( ) is a subgroup of U(n) for which the determinant equals +1. 

From any arbitrary choice of orbitals si and 38 it is possible to construct the most localized and the most delocalized orbitals through an unitary transformation. It is possible to show that the following quantities remain invariant to this transformation14 ... 

The off-diagonal elements of the matrix (3.13) contain quantities yrfa, and yrf39. Since + y2M is invariant to the unitary transformation, it is clear that for any perturbation an appropriate choice of si and 58 can make either or vanish. This will be, of course, neither the most localized nor the most delocalized orbitals. On the other hand, since has no clear physical significance, the most convenient working choice of orbitals is the one for which y m vanishes. As was already mentioned, this is the case of the most localized si = A and 58 = B and the most delocalized orbitals si = a and 58 = b. For these choices, consequently, we have to deal with two independent perturbations that are related to each other in two different basis as... 

A unitary transformation between the various orbitals of an individual atom we shall refer to this as space invariance. 

As a preliminary for deriving a useful expression, we note that the trace of a matrix is invariant under unitary transformation and that Tr(AB) = Tr (BA). For any unitary matrix 11, it follows that 0 = (LflP) = UlP + UUP and that Tr(U W + UPU) = 0, the prime indicating the energy derivative. Therefore, we have... 

One may obtain now (see [7]) the general expression of the density matrix of multi-dimensional system that is invariant to relation of the unitary transformation S ... 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

This defines the fermion contribution to an isovector gauge current density. Although the Euler-Lagrange equation is gauge covariant by construction, this fermion gauge current is not invariant, because the matrix r does not commute with the 5(7(2) unitary transformation matrices. It will be shown below that the... 

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