Singlet wave function

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... 

The wave function III. 144 has a very simple form, and applications to the He-like ions show that one can still expect a surprisingly high accuracy. Let us consider a singlet wave function of the form ... 

Before introducing the form of the Mulliken operator, w u11, it is appropriate to return to the concepts of early days of quantum mechanics. Heitler and London wrote down the singlet wave function for H2 in terms of the hydrogen Is atomic orbitals on both hydrogen atoms A and B [4] ... 

The evaluation of the transition dipole moment between the triplet and singlet wave functions considered so far... 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

We have written a combination that is antisymmetric for exchange of both space and spin coordinates. This singlet wave function can also be written... 

The condition j + j > 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule /+ / > 2 for i (2)(Eq. [171]) tells us that electronic spin-spin interaction does not contribute to their fine-structure splitting in first order. 

Because of the spin-orbit selection rules, only triplet zeroth-order states contribute to the first-order perturbation correction of a singlet wave function. In Rayleigh-Schro dinger perturbation theory, the expansion coefficient a of a triplet zeroth-order state (3spin-orbit matrix element with the electronic ground state (in the numerator) and its energy difference with respect to the latter (in the denominator). 

According to Heitler and London [1], the electronic states of the hydrogen molecule can be formed by distributing the two electrons between the atomic orbitals (AOs), Ls and 1 sb (abbreviated by a and b), on the two hydrogen atoms. The singlet wave function characterized by quantum numbers S=0 and M=0 (M denotes the z-component of total spin S) can be described using the form... 

We will present a singlet wave function which is an exact zero-energy eigenvector to each hn and, therefore, it is the exact ground-state wave function of (2). This function has a form... 

Now we calculate the norm and correlation function of the wave function To (10). The norm of the singlet wave function To is... 

It is interesting to note that the singlet wave function (26) can be also represented in a special recurrent form [14, 11]... 

The singlet wave function (60) is conveniently identified graphically with a square lattice, each site corresponding to a fourth-rank spinor (whose form is identical for all sites), and each segment linking sites corresponds to a metric spinor gXfi (Fig.8). 

For a singlet wave function (5 = 0), the number of coupling schemes for N electrons is given in Table 7.1. 

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