Unitary group formalism

We mentioned earlier that the dimensionality of the FCI space is significantly reduced due to spin symmetry. This can be formulated somewhat differently due to the relation existing between the spin and permutation symmetries of the many-electronic wave functions (see [30,42]). Indeed, the wave function of two electrons in two orbitals a and b allows for six different Slater determinants 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

These observations are valid for the wave functions of an arbitrary number of electrons. The respective generalization is done as follows first we notice that the permutation symmetry of a function is given by the so-called Young patterns figures formed by boxes, e.g.  

It is easy to figure out the relation of this with the spin of the many-electron state. It is clear that the above Young tableau corresponds to a many-electron state with the spin projection equal to 

Now we can consider the symmetry properties of the spatial functions corresponding to the above spin functions. They are uniquely defined from the requirement that the product of the spatial and spin functions must be antisymmetric. In a way, what was symmetrized in the spin part (rows) must be antisymmetrized in the spatial part (columns) and vice versa. That means that the spin function represented by a two-row Young pattern T with the first row longer by 2S boxes than the second one must be complemented by a spatial function represented by a two-column Young pattern T with the first column longer by 2S boxes than the second one, e.g.  

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Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

Scuseria, Janssen, and Schaefer, for example, developed a set of intermediates based on their reformulation of the CCSD amplitude and energy equations in a unitary group formalism designed to offer special efficiency when the refer-... 

X. Li and J. Paldus,/. Chem. Phys., 101, 8812 (1994). Automation of the Implementation of Spin-Adapted Open-Shell Coupled-Cluster Theories Relying on the Unitary Group Formalism. 

To describe the computation of the coupling coefficients with the formalism of the unitary group approach, the distinct row table (DRT) and its graphical... 

In the domain of Cl methodology, our close interaction followed the formulation of the unitary group approach (UGA) to the many-electron correlation problem by one of us [2-5]. Professor Shavitt soon fell in love with this formalism and designed an ingenious graphical representation [6-8] of the electronic Gel fand-Tsetlin (G-T) basis as described in terms of the so-called ABC or Paldus tableaux (see below). This development ultimately resulted in his graphical UGA (GUGA) version of UGA (for most recent overviews of these approaches see [4, 9-13]). 

Paunz, R., Spin Eigenfunctions. Plenum, New York, 1979. The last chapter of this monograph contains an introduction to the unitary group approach to Cl. This approach, which has recently attracted much attention, provides a formalism for the efficient computation of the Hamiltonian matrix elements between spin-adapted configurations. 

The state-selective approach to the multi-reference problem was further developed by Banerjee and Simons [118], by Laidig and Bartlett [119], by Hoffmann and Simons [120], by Li and Paldus [121, 122] and by Jeziorski, Paldus and Jankowski [123] who formulated extensive open-shell cc theory, based on the unitary group approach (uga) formalism. The Rayleigh-Schrodinger formulation of a state-selective approach to the multi-reference correlation problem has been developed more recently by Mukherjee and his collaborators [124-130] and also by Schaefer and his colleagues [131-133]. 

Unitary-group generators have in fact already been encountered in Section 8.2, where they were represented as operators in Fock space describing the mappings of a spin-orbital basis, and, although we shall use a spin-free formalism, it is also useful to see how they occur in the second-quantization approach. In (8.2.18) et seq. it was established that the operator... 

Secondly, the commutator is the Lie product33 of the operators X Xs and Xu this choice of multiplication is particularly appropriate when one realizes that the X XS are the generators of the semisimple compact Lie group U , which is associated with the infinitesimal unitary transformations of the Euclidean vector space R (e.g., the space of the creation operators).34 With the preceding comments, the action of the transformation operator on the creation operators can formally be written in the usual form of the transformation law for covariant vectors,33... 

The coordinates Si, etc., used above are illustrations of symmetry coordinates, i.e., coordinates in terms of which the secular equation is factored to the maximum extent made possible by the symmetry. In Appendix XII it is proved formally that coordinates arc symmetry coordinates if (a) they form the basis of a completely reduced unitary representation of the point group of the molecule (5) sets of coordinates of the same degenerate symmetry species have identical transformation coefficients. 

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