Spin-adapted pairs

We now investigate this invariance problem with our minimal basis model of two noninteracting H2 molecules. The following example also provides us with an opportunity to introduce the concept of spin-adapted pair correlation energies. Up to this point we have considered spin-orbital pair energies. For example, in a 4-electron system with and 2 doubly occupied, we must calculate six spin-orbital pair energies, i.e., e, 12, 12, T2> i2> 22- However, not all of the corresponding spin-orbital pair... 

Since the pair functions for the aa and tiS pairs are pure singlets the total correlation energy for the dimer using spin-adapted pairs is... 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

Computer programs were MOLCAS 3 program system (25) for SCF, CASSCF, and CASPT2 calculations and the program TITAN for closed shell calculations (26). The new version of the COMENIUS program was used for open shell CCSD(T) calculations based on the spin adapted singly and doubly excited amplitudes (15, 27-29). These codes were supplemented by the generator of the no-pair hamiltonian written by B. A. Hess in all DK calculations. 

Using all the above, Cizek presents the explicit, spin orbital and spin-adapted CC doubles equations (CCD) i.e., T = T2 (then called coupled-pair many-electron theory) in terms of one- and two-electron integrals over an orthogonal basis set. Assisted by Joe Paldus with some computations, he also reports some CCD results for N2, which though limited to only ti to Ug excitations, uses ab initio integrals in an Slater-type-orbital basis. He also does the full Cl calculation to assess convergence, a tool widely used in Cizek s and Paldus work and by most of us, today. He also reports results for the minimum-basis TT-electron approximation to benzene. 

Modern implementations of the MP2-F12 method combine the CABS approximation ]20] with robust density fitting techniques [21, 22] and local approaches ]23]. The coefficients are usually constrained at the values predetermined from the cusp conditions, as one half for singlet pairs and one quarter for triplet pairs in the spin-adapted formalism [24, 25]. The MP2-F12 methods have been extended to treat open-shell systems with unrestricted [26, 27, 28], restricted [29, 30] and multireference [29] formalisms. 

Jeziorski et al, have formulated a first-quantization form of the CCD equations where the pair functions are not expressed in terms of double replacements but as expansions in Gaussian geminals, In the original derivation of the theory, they have employed a spin-adapted formulation in terms of singlet and triplet pairs, but a spin-orbital formalism will be used in the following for the sake of a compact presentation,... 

In the spin-adapted formulation, there are n /4 pair functions, If each pair function would be expanded in N geminals, the total number of nonlinear parameters in a calculation would be equal to x (9n /4),... 

The low MW power levels conuuonly employed in TREPR spectroscopy do not require any precautions to avoid detector overload and, therefore, the fiill time development of the transient magnetization is obtained undiminished by any MW detection deadtime. (3) Standard CW EPR equipment can be used for TREPR requiring only moderate efforts to adapt the MW detection part of the spectrometer for the observation of the transient response to a pulsed light excitation with high time resolution. (4) TREPR spectroscopy proved to be a suitable teclmique for observing a variety of spin coherence phenomena, such as transient nutations [16], quantum beats [17] and nuclear modulations [18], that have been usefi.il to interpret EPR data on light-mduced spm-correlated radical pairs. 

Redress can be obtained by the electron localization function (ELF). It decomposes the electron density spatially into regions that correspond to the notion of electron pairs, and its results are compatible with the valence shell electron-pair repulsion theory. An electron has a certain electron density p, (x, y, z) at a site x, y, z this can be calculated with quantum mechanics. Take a small, spherical volume element AV around this site. The product nY(x, y, z) = p, (x, y, z)AV corresponds to the number of electrons in this volume element. For a given number of electrons the size of the sphere AV adapts itself to the electron density. For this given number of electrons one can calculate the probability w(x, y, z) of finding a second electron with the same spin within this very volume element. According to the Pauli principle this electron must belong to another electron pair. The electron localization function is defined with the aid of this probability ... 

Eq. (1) strictly only applies to molecules and complex ions and ignores perturbations arising through cooperative effects in binary compounds Ly in which the coordination sphere ML is indiscrete. It should be noted that in obtaining the two sets of j,e.t. jjjg value for each metal ion is adapted so that the effects of spin pairing and lODq are taken into account for our purposes, we require the uncorrected frequency rather than PeoJiected (s Appendix). 

Fig. 5. Electron spin pairing in metal-ammonia solutions at 25°C (298 K), 0°C (273 K), and -33°C (240 K). Paramagnetic spin concentrations for sodium-ammonia and potassium-ammonia solutions. [Experimental data from Refs. 47, 76, 98, 114,115, and 159. Adapted from Harris (88). Used with permission.] The solid line indicates the expected spin-pairing behavior for noninteracting electrons (88).

Figure 3.16 Energy level diagram for ferric iron matched to spin-forbidden crystal field transitions within Fe3+ ions, which are portrayed by the polarized absorption spectra of yellow sapphire (adapted from Ferguson Fielding, 1972 Sherman, 1985a). Note that the unassigned band at -17,600 cm-1 represents a paired transition within magnetically coupled Fe3+ ions located in adjacent face-shared octahedra in the corundum structure.

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