Projection wave function

As one can see, the operator has a property of the wave operator (it transforms the projection of the exact wave function into the exact wave function), however, it should be stressed that the operator converts just one projected wave function into the corresponding exact wave function so we will denote it as a state-specific wave operator in contrast to the so-called Bloch wave operator [46] that transforms all d projections into corresponding exact states. From definition (11) it is iimnediately seen that the state-specific wave operators obey the following system of equations for a = 1,..., d... 

Such a wave function was used for the two-reference MR BWCCSD calculation of E2 in Section 18.2, where was the (l7Tg) (3crg) configuration and reference configurations has been employed in the study of IBr [56] and oxygen molecule [57]. The superscript P is used to indicate that P q a projected wave function. Its relation to the exact wave function Eq is provided by the (so far) unknown wave operator ... 

In the first place, the projected wave functions have to be expressed in terms of the 3d orbitals by substituting the expressions of multideterminantal wave functions P. This gives... 

This operator acts within the model space and will be referred to as a multi-root effective Hamiltonian operator. The eigenfunctions of the effective Hamiltonian operator are projected wave functions Since the functions are not necessarily orthogonal, the effective Hamiltonian operator, TCeff, is, in general, non-Hermitian. Clearly, the eigenvalues of Tfeff are always real since they correspond to true energies. 

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

The initial amplitudes d/(to) are obtained by projecting the initial wave function on the DVR basis set. For the initial wave function, we use... 

The projection on the final channel is done in the following manner. We let the trajectory decide on the channel—just as in an ordinary classical trajectory program. Once the channel is detemrined we project the wave function (in the DVR representation) on the appropriate wave function for that channel... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

Another approach is to run an unrestricted calculation and then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). This gives a correction to the energy but does not affect the wave function. Spin projection nearly always improves ah initio results, but may seriously harm the accuracy of DFT results. 

If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more significant, use restricted open-shell methods. If all else fails, use highly correlated methods. 

Many transition metal systems are open-shell systems. Due to the presence of low-energy excited states, it is very common to experience problems with spin contamination of unrestricted wave functions. Quite often, spin projection and annihilation techniques are not sufficient to correct the large amount of spin contamination. Because of this, restricted open-shell calculations are more reliable than unrestricted calculations for metal system. Spin contamination is discussed in Chapter 27. 

Analogously to MP methods, coupled cluster theory may also be based on a UFIF reference wave function. The resulting UCC methods again suffer from spin contamination of the underlying UHF, but the infinite nature of coupled cluster methods is substantially better at reducing spin contamination relative to UMP. Projection methods analogous to those of the PUMP case have been considered but are not commonly used. ROHF based coupled cluster methods have also been proposed, but appear to give results very similar to UCC, especially at the CCSD(T) level. 

At the dissociation limit the UHF wave function is essentially an equal mixture of a singlet and a triplet state, as discussed in Section 4.4. Removal of the triplet state by projection (PUHF) lowers the energy in the intermediate range, but has no effect when the bond is completely broken, since the singlet and triplet states are degenerate here. 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

If the wave function W has a symmetry property characterized by a projection operator 0, the expansion IV. 1 may be replaced by the series... 

Fieschi, R., and Lowdin, P.-O., "Atomic state wave functions, generated by projection operators."... 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

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