Total spin

In UHF theory, the expectation value of the total spin operator over the single-determinantal UHF wave function is computed as 

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. 

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An atom or a molecule with the total spin of the electrons S = 1 is said to be in a triplet state. The multiplicity of such a state is (2.S +1)=3. Triplet systems occur in both excited and ground state molecules, in some compounds containing transition metal ions, in radical pair systems, and in some defects in solids. 

The identical colliding particles, each with spin s, are in a resolved state with total spinin the range (0 2s). The spatial wavefiinction with respect to particle interchange satisfies = (—1 Wavefunctions for identical particles with even or odd total spin. S are therefore symmetric (S) or antisynnnetric (A) with respect to particle... 

Let and be the fractions of states with odd and even total spins. Sj =0,1,2,.. ., 2s. When the 2s + 1 spin-states. Sj are iimesolved, the appropriate combination of syimnetric and antisyimnetric cross sections is the weighted mean... 

You can also plot ihe electrostatic polenlial. the total charge density. or the total spin density determined during a semi-enipincal or ah initio calculation. This information is useful in determining reactivity and correlating calculalional results with experimental data. Th ese examples illustrate uses of lb ese plots ... 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

Secondly, you must describe the electron spin state of the system to be calcn lated. Electron s with their individual spin s of Sj=l /2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin... 

If the mini her of electrons, N, is even, yon can haven dosed shell (as shown ) where the occupied orbitals each contain two electron s. For an odd n nrn her of electron s, at least on e orbital rn ust be singly occupied. In the example, three orbitals are occupied by-electron s and two orbitals arc nn occupied. Th e h ighest occupied nioleciilar orbital (HOMO is t[r), and the lowest unoccupied molecular orbital (LUMO) is The example above is a singlet, a state oh total spin S=0. Exciting one electron from the HOMO to the LUMO orbital would give one ol the I ollowing excited states ... 

Total spin S=0 (a singlet) if the spins remained the same. 

Total spin S=1 (a triplet) il a spin-down electron in the HOMO was promoted to be a spm-iip electron in the l.LMO. That is, the spin of one oTthe electrons was reversed when exciting it from the HOMO to the LUMCL... 

If yon add a single electron to the LUMO orbital above to create an anion, you obtain total spin S=l/2 (a donhlet). 

One identifies the highest Ms value (this gives a value of the total spin quantum number that arises, S) in the box. For the above example, the answer is S = 1. 

The advantage of unrestricted calculations is that they can be performed very efficiently. The alpha and beta orbitals should be slightly different, an effect called spin polarization. The disadvantage is that the wave function is no longer an eigenfunction of the total spin <(5 >. Thus, some error may be introduced into the calculation. This error is called spin contamination and it can be considered as having too much spin polarization. 

As a check for the presence of spin contamination, most ah initio programs will print out the expectation value of the total spin <(A >. If there is no spin contamination, this should equal. v(.v + 1), where s equals times the number of unpaired electrons. One rule of thumb, which was derived from experience with... 

In addition to total energy and gradient, HyperChem can use quantum mechanical methods to calculate several other properties. The properties include the dipole moment, total electron density, total spin density, electrostatic potential, heats of formation, orbital energy levels, vibrational normal modes and frequencies, infrared spectrum intensities, and ultraviolet-visible spectrum frequencies and intensities. The HyperChem log file includes energy, gradient, and dipole values, while HIN files store atomic charge values. 

Secondly, you must describe the electron spin state of the system to be calculated. Electrons with their individual spins of sj=l/2 can combine in various ways to lead to a state of given total spin. The second input quantity needed is a description of the total spin S=Esj. Since spin is a vector, there are various ways of combining individual spins, but the net result is that a molecule can have spin S of 0, 1/2, 1,. These states have a multiplicity of 2S-tl = 1, 2, 3,. ..,that is, there is only one way of orienting a spin of 0, two ways of orienting a spin of 1/2, three ways of orienting a spin of 1, and so on. 

If all spins ( 1/2) in an atom or molecule are paired (equal numbers of spin +1/2 and -1/2), the total spin must be zero, and that state is described as a singlet (total spin, S = 0 and the state is described by the term 2S + 1 = 1). When a singlet ground-state atom or molecule absorbs a photon, a valence electron of spin 1/2 moves to a higher energy level but maintains the same... 

Figure 7.4 Russell-Saunders coupling of (a) orbital angular momenta li and I2, (b) spin angular momenta Sj and 2 and (c) total orbital and total spin angular momenta, L and S, of sip and a d electron...

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

The spin multiplicity for a molecule is given by the equation 2S + 1, where S is the total spin for the molecule. Paired electrons contribute nothing to this quantity. They have a net spin of zero since an alpha electron has a spin of +Vi and a beta electron has a spin of -Vi. Each unpaired electron contributes +Vi to S. Thus, a singlet—a system with no unpaired electrons—has a spin multiplicity of 1, a doublet (one unpaired electron) has a spin multiplicity of 2, a triplet (two unpaired electrons of like spin) has a spin multiplicity of 3, and so on. 

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