Symmetric tensor product

For any natural number n, define the symmetric tensor product... 

The definitions of the perturbation corrections to the interaction energy, as given by Eqs. (8), (9), and (11), involve nonsymmetric operators, like H0, V, and Rq. These operators do not include all electrons in a fully symmetric way, so H0, V, and Rq must act in a larger space them the dimer Hilbert space Hab adapted to a specific irrep of 4. To use these operators we have to consider the space Ha Hb, the tensor product of Hilbert spaces Ha and Hb for the monomers A and B. For the interaction of two closed-shell two-electron systems this tensor product space should be adapted to the irreps of the 2 2 group. In most of the quantum chemical applications the Hilbert spaces Hx, X = A and B, are constructed from one-electron spaces of finite dimension Cx - In the particular case of two-electron systems in singlet states we have Hx — x x, where the symbol denotes the symmetrized tensor product. We define the one-electron space as the space spanned by the union of two atomic bases associated with the monomers in the dimer. The basis of the space includes functions centered on all atoms in the dimer and, consequently, will be referred to as the dimer-centered basis. We assume that the same one-electron space is used to construct the Hilbert spaces Hx 1 X = A and B, i.e., Hx = In such case Ha )Hb can be represented as a direct sum of Hilbert spaces H adapted to the irreps entering the induced product [2] [2] 4, i.e., Ha ( Hb = H[2i] f[3i] f[4]- This means that every function from H , v = [22], [31], or [4], can be expanded in terms of functions from Ha Hb-... 

As in a Cartesian coordinate system the tensor product u v, of the vectors u and v, has the elements uxvp, the EFG tensor is a symmetric tensor with elements... 

The product of any 3N vector with this geometric projection tensor isolates the soft component of that vector. The geometrical projection tensor is a symmetric tensor, like the Euclidean identity and unlike the dynamical projection tensor. To reflect this fact, its bead indices are written directly above and below one another, with no offset to indicate whether the implicit Cartesian index associated with each bead index acts to the right or left. 

The " " represent scalar products and the "x" represent tensor products. The purpose of the Majorana term was to remove states which are non-symmetric under interchange of the proton and neutron degrees of freedom by shifting them up in energy. The non-symmetric neutron-proton states are now of some physical interest [BOH 84]. For simplicity, and to decrease the number of variable parameters, we have taken = ev = e. In general, this will not be true. 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

The vector and tensor differential operators that have been applied to the symmetrical dyad product pvv can be defined (e.g., [12], p. 574) ... 

The latter result (82) yields a quantum probability amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate downconversion. To see this, we note that the matrix element for SHG invokes the tensor product Py (—2co co, ) p([/lC., where the brackets embracing two of the subscripts (jk) in the radiation tensor denote index symmetry, reflecting the equivalence of the two input photons. As shown previously [1], this allows the tensor product to be written without loss of generality as ( 2co co, co), entailing an index-symmetrized form of the molecular response tensor,... 

The tensor product Tik m represents, in first approximation, the field created by molecule k in the region occupied by the molecule The tensor T is a symmetrical dyad defined by... 

Axiality of w is automatically achieved by the usual transformation ((c) in Rem. 4) of tensor W. Therefore the skew-symmetric tensors instead of axial vectors and outer product (see Rem. 16) may be used and we do it this way at the moment of momentum balances in the Sects. 3.3,4.3, cf. [7, 8, 14, 27]. Generalization of this Lemma to third-order tensors, made by M. Silhav, is published in Appendix of [28]. 

We use the outer product A defined for two vectors a, basaAb = a b — b a, i.e. (a A h)d = a b-i — a-ib . This product is obviously the skew-symmetric tensor which, using the results from Rem. 10, is equivalent to the axial vector created by the vector product of these vectors, see Rem. 6... 

The fictitious spin operator indeed transforms as a Ti operator and has the tensorial rank of a p-orbital. However, as we have shown, the full Hamiltonian also includes a Jf part, which involves an /-like operator. To mimic this part by a spin Hamiltonian, one thus will need a symmetrized triple product of the fictitious spin, which will embody an /-tensor, transforming in the octahedral symmetry as the T irrep. These /-functions can be found in Table 7.1 and are of type z 5z - ir ). But beware To find the corresponding spin operator, it is not sufficient simply to substitute the Cartesian variables by the corresponding spinor components, i.e., z by 5j , etc. indeed, while products of x, y, and z are commutative, the products of the corresponding operators are not. Hence, when constmcting the octupolar product... 

The next step is to realise that symmetric tensors can be constmcted from two vectors. In matrix algebra, this is done by forming the outer product of the two vectors. Thus any symmetric tensor f can be constmcted from the proper choice of the two vectors A and B... 

We can define other deformation tensors, also, in terms of the deformation gradient tensor F. According to the polar decomposition theorem of the second-order tensor (see Appendix 2A), the deformation gradient tensor F, which is an asymmetric tensor and is assumed to be nonsingular (i.e., det F 0), can be expressed as a product of a positive symmetric tensor with an orthogonal tensor (Jaunzemis 1967) ... 

Because the scalar inner product of a symmetric and an antisymmetric tensor vanishes, from (A.l 1)... 

The polarizability tensor of a molecule related the components of the induced dipole moment of the molecule to the components of the electric field doing the inducing. It therefore has 9 components, axx, ctxy, etc., only 6 of which are independent. The theory of the Raman effect shows that a vibrational transition, from the totally symmetric ground state to an excited state of symmetry species F, will he Raman active if at least one of the following direct products contains the totally symmetric representation ... 

This expression must be independent of the choice of the co-ordinate system. In line with the general usage to look upon the 6 independent elements of the symmetric strain (stress) tensor as components of a vector, we may elucidate this independence of co-ordinate system by writing eq. (2) as the scalar product of the vectors [fijrjn yy< Exyyj2., vj /2, and [/l2, fl2 3 > 2 /2, fafa-J2, faP -y/2]. The... 

For tensors of higher rank we must ensure that the bases are properly normalized and remain so under the unitary transformations that correspond to proper or improper rotations. For a symmetric T(2) the six independent components transform like binary products. There is only one way of writing xx xx, but since xx x2 = x2 xx the factors xx and x2 may be combined in two equivalent ways. For the bases to remain normalized under unitary transformations the square of the normalization factor N for each tensor component is the number of combinations of the suffices in that particular product. F or binary products of two unlike factors this number is two (namely ij and ji) and so N2 = 2 and x, x appears as /2x,- Xj. The properly normalized orthogonal basis transforming like... 

Therein, T" = (T )r are the symmetric partial Cauchy stress tensors, // b represent the body forces and p" are the momentum productions, where ps + pF = o must hold due the overall conservation of momentum. 

The advantage of the symmetrized SF tensor is that, since the two sides of equation (51) must transform in the same way under all point group operations, the element will be non-zero only if the product of co-ordinates QrQtQt... transforms in the same way as the co-ordinate Si. 

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