Pure spin states

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

A major problem with the UHF method is that it does not give a pure spin state, so that... 

The amount of spin contamination is given by the expectation value of die operator, (S ). The theoretical value for a pure spin state is S S + 1), i.e. 0 for a singlet (Sz = 0), 0.75 for a doublet (S = 1/2), 2.00 for a triplet (S = 1) etc. A UHF singlet wave function will contain some amounts of triplet, quintet etc. states, increasing the (S ) value from its theoretical value of zero for a pure spin state. Similarly, a UHF doublet wave function will contain some amounts of quartet, sextet etc. states. Usually the contribution from the next higher spin state from the desired is... 

A single Slater determinant with N+ N represents a pure spin state if, and only if, the number of doubly filled orbitals defined by Eq. 11.57 equals AL. 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

Leaving the question of pure spin states entirely aside, Wigner studied a system containing an even number of electrons (N = 2n) by considering the product of two determinants built up from or-bitals only... 

Wigner s study of the correlation effects for high-electron densities is closely connected to the standard methods described in Section III.E., and the main errors come from the restricted form of the wave function (Eq. III.7) and the fact that this function does not represent a pure spin state. Hence, Wigner obtains only an upper bound for the correlation energy in this case. 

As shown in Section II.D(2), the determinant of Eq. III.133 can be brought to correspond to a pure spin state by imposing a certain condition (11.61) on the relation between p+ and p. which corresponds to the pairing of the electrons. If p+ and p are permitted to vary independently of each other, the determinant is no longer a pure spin state but a mixture of states associated with the quantum numbers... 

It is now possible to formulate an extension of the conventional Hartree-Fock scheme by considering a wave function (25+1) IP which is a pure spin state and which is simply defined by the component of the single Slater determinant Eq. III. 133 as has the spin property required ... 

Pure spin states with Hartree-Fock method, 227, 230... 

Since in those forms of the UHF wave functions, one drops a constraint (either the need of a pure spin state in the first case or the Pauli antisymmetry rule in the second case), it is expected that the resulting wave function will give a lower energy than in the RHF case and thus introduce a part of the correlation energy. As shown in the table above, there is... 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

The actual eigenstates are equal admixtures of the two unperturbed pure spin states when the field is exactly at the value at which the crossing would have occurred (v,m = 0). Since initially (when the muon stops) the system is in a well defined muon spin state, i.e., one of the two unperturbed pure spin states, the system oscillates at the frequency vT between the muon spin being along and opposite to the field, as implied by Eqs. 10 and 11. Thus, upon time averaging the positron counts the forward-backward asymmetry is reduced. 

Spin-restricted procedures, signified by an R prefix (e.g. RHF, RMP), constrain the a and (3 orbitals to be the same. As such, the resulting wavefunctions are eigenfunctions of the spin-squared operator (S2) that correspond to pure spin states (doublets, triplets, etc). The disadvantage of this approach is that it restricts the flexibility in the... 

Rather loosely, this may be thought of as taking U ao, so that double occupancy of 0) is totally forbidden. Because the wavefunction (11) has different a and /9-spin spatial orbitals, it is not a pure spin state (it will be a mixture of singlet, triplet, quintet, etc.). However, for the problem in hand, this does not seem to be much of a disadvantage. 

First-order perturbation theory is then apphed to derive the nominal "singlet ground state and first excited triplet functions. Pure spin states are no longer possible. 

It is evident that a 2-RDM that corresponds to a Hamiltonian eigenstate also corresponds to a pure-spin state. However, when one is working with an approximated RDM, it is important that this RDM should correspond to a spin eigenstate. 

D. R. Alcoba and C. Valdemoro, Spin structure and properties of the correlation matrices corresponding to pure spin states controlling the S-representability of these matrices. Int. J. Quantum Chem. 102, 629 (2005). 

Furthermore, properties of the spin components of the 2-G can be obtained by reconsidering the spin properties of the 1-TRDMs. Thus the different spin-blocks of the 1-TRDMs can be related among themselves through the action of the operator S on pure spin states. One therefore has... 

Similarly, application of the properties of the spin-shifting operators, S , allows one to obtain the relations connecting the 1-TRDMs corresponding to different multiplet states. Thus, by considering the action of the spin-shifting operator S+ on pure spin states,... 

The above relations, which are represented in a spin-orbital basis, are especially relevant they are analytical results that describe all the conditions that a 2-G corresponding to any pure spin state must satisfy hence they constitute a complete set of S -representability conditions. Their generality imphes a general usefulness within the framework of any RDM methodology. 

Hence the general relations addressed in Section II are also valid for the 2-CM. Thus the different spin components of a 2-CM are also related when this matrix corresponds to a pure spin state with spin quantum numbers S and M and similarly for the 2-CMs corresponding to different states of a given multiplet. 

If the system is in a pure spin state, that is, = 5(5-f 1) and 5 = Mj F, then the electron density can be decomposed into spin components as follows... 

The ordinary unrestricted Hartree-Fock (UHF) function is not written like either of these. It is not a pure spin state (doublet) as are these functions. The spin coupled VB (SCVB) function is lower in energy than the UHF in the same basis. 

We wish to apply permutations, and the antisymmetrizer to products of spin-orbitals that provide a basis for a variational calculation. If each of these represents a pure spin state, the function may be factored into a spatial and a spin part. Therefore, the whole product, 4, may be written as a product of a separate spatial function and a spin function. Each of these is, of course, a product of spatial or spin functions of the individual particles,... 

Consider an n electron system in a pure spin state S. The associated partition is n/2 + S,n/2 — S], and the first standard tableau is... 

In DFT, there is no formal way to evaluate spin contamination for the (unknown) interacting wave function. As has already been discussed in Sections 8.5.1 and 8.5.3, however, the expectation value of S - computed from Fq. (9.30) over the KS determinant can nevertheless sometimes provide qualitative information about the likely utihty of the DFT results with respect to their interpretation as corresponding to a pure spin state compared to a mixture of different spin states. 

But, as described in more detail in Appendix C, this wave function, which configurationally corresponds to an a electron in the orbital and a p electron in the ( 2 orbital, is neither a singlet nor a triplet, but a 50 50 mixture of the two, and this point is emphasized by die left superscript on 4/ in Eq. (14.14). While the wave function does not represent a pure spin state, we may take advantage of the prevailing situation by noting that we may write... 

Finally, consider the case where die overlap between the a and fi orbitals is exactly zero (which could happen, for instance, if all a MOs were on one atom and all MOs on another atom with the two atoms infinitely far apart). In that case, the expectation value will be larger than the pure spin state where only the excess a electrons are unpaired, but smaller than the value expected for the pure spin state where all electrons are unpaired (i.e., a low-spin (n-l-l)-multiplet where n is the total number of electrons, for which the expectation value would be computed using s = ( a +np)/2 instead of j = ( — n/i)/2). Such a system is said to be spin-contaminated because it is a mixture of die lowest spin state and varying contributions from states of higher spin multiplicity. Obviously, such wave functions are of limited utility, since expectation values of other properties will also represent an admixture of the properties of the different states. 

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