Spin expectation value

A further idea to improve Yamaguchi s expression was introduced by Clark and Davidson (113,114) who suggested to replace total spins with local spin expectation values of surrogate spin operators localized at atomic centers,... 

It can be evaluated by calculation of its expectation value. For this purpose, we need to calculate the exchange integral Kab as given by Clark and Davidson in Eq. (84) and then evaluate the local spin expectation values for the operators SA SA. 

In order to present the derivation of local spin expectation values, we shall briefly recall the foundations of spin eigenvalue equations in the non-relativistic framework. Information on the spin states of a molecule can be extracted from either the total spin operator S2 or its -component Sz (i.e., from its projection on the -axis),... 

For the total spin expectation value several definitions exist. Here, we choose Lowdin s representation (128)... 

A detailed analysis of (S2) in DFT can be found in Refs. (129,130).) Note the change of meaning in the summation indices in Eq. (91) In the second line i,j label electronic (spin) coordinates, while they denote the indices of spin orbitals in the third line. For the study of open-shell transition-metal clusters, it is necessary to obtain an expression for the total spin expectation value, where the summation rims over the number of a- and / -electrons rather than over the total number of electrons N. Thus, the sum in Eq. (91) may be split into four sums over the various spin combinations,... 

In order to obtain expressions for the local spin expectation values, different decomposition schemes exist. One may either partition the total spin expectation value (S2) (122,124) as suggested by Mayer or the total spin operator S2 (113,114) as proposed by Clark and Davidson. The corresponding decomposition schemes for multi-determinant wave functions may be found in Refs. (125-127). 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

Knowing both decomposition schemes yield the same squared total spin expectation value, we may write the expectation value of the total spin operator S2 in terms of Mayer s local expressions... 

Selected Ideal Total and Local Spin Expectation Values for the [Fe2] Cluster 1, Where Each Metal Center Carries Four Unpaired Electrons Resulting in a Total Spin State of S=4 for the High-Spin State. Local Spin Values on the Iron Atoms are Displayed in the Last Column. Up-Pointing Arrows Indicate Excess of q-Spin and Down-Pointing Arrows Excess of /3-Spin... 

Table II up- and down-pointing arrows were chosen to illustrate the local spin distributions, where an up-pointing arrow represents an electron with -spin and a down-pointing arrow an electron with / -spin. The ideal total spin expectation values are given in terms of the eigenvalues and local spins on the metal centers are indicated by ideal Msxvalues.

Note that in Table III local (Sza), (S ), and (Sa Sb) expectation values for the [Fe2] cluster 1 are listed. As shown in Section V the local expressions for the Mayer and the Clark and Davidson decomposition schemes differ. Hence, different values for the (S ), e.g., for partitioning schemes. The same argument holds for the cross terms (Sa Sb), here (Spei Sk< 2)- The total spin expectation value (S2) is shown to illustrate that spin contamination is fairly low for most spin states. The reader should not be confused by the fact that summation over the local spin values at the iron centers (Spel), (Spe2>, and (Spei SFe2) does not yield the same value for Mayer and for Clark and Davidson data and that they do not sum up to (S2). The total spin expectation value (S2) is only obtained when summing all local spin values of all atoms of the [Fe2] cluster 1. 

The Total Spin Expectation Value S2 and Selected Local (SzA), (Sa), and (Sa SB) Values of the [Fe2] Cluster 1 Obtained with Mulliken Projection Operators are Given for Local Decomposition Schemes According to Mayer and to Clark and Davidson for Single-Point Calculations on the BP86/M/TZVP Optimized High-Spin Structure... 

Once these eigenstates have been determined, the individual site spin expectation values can be obtained as... 

The method described above is of general validity and can be applied to transition metal clusters of arbitrary shape, size, and nucleanty. It should be noted that in the specific case of a system comprising only two interacting exchanged coupled centers, our general treatment yields the same result as that of Bencini and Gatteschi (121), which was specifically formulated for dimers. In this case, the relation between the spin-projection coefficient and the on-site spin expectation value is simply given by... 

Note that the second and third integrals on the r.h.s. are zero because of the orthonormality of the spatial orbitals a and b, whose products appear over the same electronic coordinate in those integrals. The spatial functions integrate to one in the first and fourth integrals, and the remaining spin expectation values are just diose of Eq. (C.17). Thus, the expectation value of Eq. (C.23) is (1 — 0 — 0+1) = 1. With additional work, it can be shown that 50 50q, not an eigenfunction of S-. 

Kramers theorem requires that all half-integer spin systems be at least doubly degenerate in the absence of a magnetic held. Next, note that the splitting of these levels by a magnetic held depends on its orientation relative to the axes of the ZFS tensor of the metal ion. The VTVH MCD saturation magnetization curve behavior reflects the difference in the population of these levels and their spin expectation values in a specific molecular direchon. This direction must be perpendicular to the polarizations of the transition (Mih where i / j are the two perpendicular polarizations... 

Increasing the field, Zeeman both splits the doublets (by g/3H cos 8), where 8 is the angle of the magnetic field relative to the molecular z-axis, and changes the wavefunctions such that the wavefunction of the lowest level goes from ( 2) — — 2)) at zero field, which is MCD inactive, to the pure —2), which will have a large MCD signal based on its spin expectation value. 

FIGURE 2.4 Spin expectation values of the lowest spin level of the 5 = 2 system of Figure 2.3. The curve labeled (Sy) was obtained by applying B along the y-axis of the ZFS tensor. The break in the curve for (Sz) is due to level crossing for BZ = 4.6T, the first excited state becomes the ground state. This diagram is relevant for the diiron(IV) complex of Section. ... 

Answer Spin expectation values (and therefore the magnetic hyperfine field) of half-integer spin systems can be saturated in low fields see Section 3.5.2). This is not the case here. In order to drive the magnetic splitting to its saturation, fields of up to 8 T have been applied. In addition, J is EPR silent, which points to an integer-spin species. 

Although the spin state of atom (or a group of atoms) A within a molecule is not observable, local spins are employed for the description of spin-spin interactions between magnetic centers, similar to the metal centers of transition-metal clusters, in terms of a Heisenberg spin-coupling model and led to considerable interest in the development of partitioning schemes of the total spin expectation value during the past decade [20, 112-128]. 

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