Direct square

The redox properties of a series of heterometal clusters were assessed by electrochemical and FPR measurements. The redox potentials of derivatives formed in D. gigas Fdll were measured by direct square wave voltammetry promoted by Mg(II) at a vitreous carbon electrode, and the following values were determined 495, 420,... 

The radial velocity is seen to be directly proportional to the temperature driving force and inversely proportional to the square root of the elapsed time of growth. On the other hand the bubble radius is proportional to the temperature driving force and to the direct square root of the time. The interesting result is that the product of the bubble radius and its radial velocity is independent of time. 

Group theory predicts that time-odd interactions for double-valued representations must transform as the symmetrized direct square. For E and E" representations one has ... 

Note in the second line of this equation that symmetrization of the direct square gives rise to only one cross-term. This equation expresses the standard Jahn-Teller result that time-even interactions in a degeneracy space transform according to the symmetrized square (indicated by square brackets) of the corresponding irrep. This square can be further resolved, into a non-distortive totally symmetric part and the proper Jahn-Teller part. 

On the other hand, if the coupling coefficients are antisymmetric under exchange of the labels, the coupled state belongs to the antisymmetrized direct square, denoted as fa. This product space is restricted to combinations with ya yb, its dimension is equal to (n —1)/2. The characters for either part of the square can be determined separately. For the character of the [Pa part the derivation runs as follows one first applies a symmetry operator to an arbitrary antisymmetric function. The ket product Paya l)) Payb(2)) will be abbreviated here as ya(l)yh(2). 

Note that this product contains one totally-symmetric irrep, notably in the symmetrized part. In general, for irreps with real characters the totally-symmetric irrep, Pq, appears in a direct square only once. This can easily be derived from Eq. (6.8). When Pa is an irrep with real characters, one has ... 

Eq. (6.22) then leads to the conclusion that the unique totally-symmetric irrep belongs to the symmetrized part of the direct square ... 

The situation is different when coupling two equivalent electrons these are electrons that belong to the same shell. In this case, the coupled states are already eigenfunctions of the exchange operator as a result of the special symmetrization properties of the coupling coefficients for direct squares. Equation (6.10) will take the following form ... 

The coupling coefficient on the right-hand side of Eq- (6.57) restricts the symmetry of the nuclear displacements to the direct square of the irrep of the electronic wave-function. This selection rule is made even more stringent by time-reversal symmetry. The Hamiltonian is based on displacement of nuclear charges, and not on momenta, so as an operator it is time-even or real. For spatially-degenerate irreps, which are... 

The transformations of the standard vector form the fundamental irrep of spherical symmetry. All other irreps can be constructed by taking direct products of this vector. In particular, the spherical harmonic functions can be constructed by taking fully symmetrized powers of the vector. The symmetrized direct square of the / -functions yields a six-dimensional function space with components z, x, yz, xz, xy. This space is not orthonormal the components are not normalized, and the first... 

The octahedral double group contains a four-dimensional spin representation, which is commonly denoted as the Tg quartet, or U in Griffith s notation. The direct square of this irrep is given by... 

For 4Jp + 2Jf = 0, the matrix splits into two separate 2x2 blocks, which have the same eigenvalues. The splitting pattern is thus as in the central panel of Fig. 7.5. Such a case can occur for a state. The orbital part of this state has no angular momentum, since the corresponding operator is not included in the direct square Ti E x E. As result, the magnetic moment of such a state is due only to the doublet spin part. Such a state behaves as a pseudo-doublet. 

Extensive direct-product tables are provided by Herzberg [10]. Antisymmetrized and symmetrized parts of direct squares are indicated by braces and brackets, respectively. 

The magnetic dipole operator transforms as T g, while the direct square of eg irreps yields A g + A2g + Eg. Since the operator irrep is not contained in the product space, the selection rules will not allow a dipole matrix element between Cg orbitals. 

The direct square of the e-irrep in D2d yields four coupled states ... 

In X-ray spectroscopy, detectabilities are a direct square-root function of analysis time. Some indication of lower analysis limits can be mentioned,... 

The redox properties of a series of hetero metal clusters were accessed by electrochemical mewasurements and EPR. The derivatives formed in D.gigas Fdll were measured by direct square wave voltammetry promoted by Mg(li) at a vitreous carbon electrode and the following redox transitions were delected Cd(-495 mV), Fe(-420 mV), NI(-360 mV) and Co(-245 mV) [vs NHE]. The values aggree well with independent measurements ((51) and our unpublished results). Similar derivatives generated in D.africanus Fdlil measured by cyclic voltammetry in film and bulk solution yield Cd(-590 mV), Zn(-490 mV) and Fe(-400 mV) (48,52). A novel addition to these series was... 

As we have shown elsewhere, the coefficients in this expression correspond to the elements of the bond-order matrix [4]. Because the JT Hamiltonian is Hermitian and invariant under time reversal, it is represented by a symmetric matrix in a real function space. The symmetries of this interaction matrix will therefore correspond to the symmetrized direct square of the degenerate irrep of the function space. This part of the direct square is represented by square brackets. The corresponding character is given by [8] ... 

In Table 4, we show how this applies to the direct square of the G representation. It reduces as follows ... 

Table 4 Derivation of the characters for the symmetrized direct square [G G] in S5...

---