Operator space basis

In order to evaluate the spectral density of Eq. (35) or (38), one needs a complete basis set spanning the lattice operator space. This basis set can be obtained by taking direct products of Wigner rotation matrices,... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

The terms variant and covariant refer to the transformation properties of the quantities. A transformation may be defined by the transformation matrix T operating on the direct space basis a, such that... 

It appears that the three-space basis associated with each spin vector operator is arbitrary. Thus operator opS (and/or opI) need not necessarily be taken as quantized along some obvious physical direction, such as that of an applied magnetic field B. In other words, spin operators opS and opI need not be expressed in the same space, that is to be quantized along the same spatial directions (i.e. the spin projection quantum numbers may be measured along different selected directions in our three-space). The most general case, which occurs when the two quantization axes are not aligned, prevents the parameter matrices from being tensors. 

A completely different approach in trying to calculate the eigenvalues of the Liouvillian is rendered by the equation-of-motion method, which was developed in nuclear physics12 13 and later introduced into quantum chemistry by several research groups.19 The basic idea is to try to solve the eigenvalue problem (1.22) by expanding the approximate eigenele-ment D in terms of a truncated basis B = Br of order m in the operator space, so that... 

Goscinski et al. applied boldly this inner projection technique to the superoperator space in order to obtain an approximation k(z) to the superresolvent R(z) in terms of the CBP and the basis B = , B2,. . . , Bm in the operator space according to the formula ... 

One may utilize this fact to construct approximate solutions to the eigenvalue problem (1.22) for the Liouvillian. For this purpose, we will introduce a set B = B, Bi,. . . , Bm of m linearly independent HS operators in our operator space, and we will then try to expand the approximate eigenoperator D in this truncated basis, so that... 

Starting from an approximate eigenelement D = Bd expressed in terms of the truncated basis B in the operator space, it is hence always possible to derive a new eigenelement D(X) which automatically satisfies the algebraic conditions (1.32). 

It should be observed, however, that one cannot in general expect that an arbitrarily chosen basis B in the operator space should satisfy these three conditions, and this applies particularly to the particle-hole operator bases commonly used in the special propagator methods or the EOM approach. In such a case, the new operator basis B(1) defined by the linearly independent elements in the four matrices (2.31) may serve as a new starting basis. Due to the construction, the basis B(,) is automatically closed under adjunction (t). It is also clear that the basis B(l) is closed under multiplication due to the fact that one has the multiplication rule... 

T = d>)( for the operator for the reference state. Starting from a given basis B in the operator space, we will further assume that we have constructed an approximate eigenoperator D = Bd by properly solving the double-commutator equations (1.49) and (1.50). The approximate unnormalized wave function for the final state is then given by the relation ... 

B.7). It is hence clear that the entire problem will be essentially simplified if the set B contains T and is closed under adjunction (t) and multiplication. If not, one has to proceed as in the previous subsection and construct a new basis B(1) in the operator space for which the relation (2.36) is going to be valid. 

In the previous discussion, it was assumed that we could start from any truncated basis B = B, B, . . . , Bm consisting of m linearly independent elements in the operator space. In this subsection, we will instead use a very special operator basis constructed by means of ket-bra operators ft)(a in general, such operators have turned out to be very useful tools in going from a carrier space with a binary product (a b) to the associated operator space. 

If

truncated basis of order n in the carrier space, then the corresponding basis P = Pk in the operator space is of order m = n2. It may further be shown4 that, if n —> °° and the basis tp = < becomes complete in the carrier space, then the basis P = Pki becomes complete in the HS operator space. It should perhaps be observed that this completeness theorem is somewhat different in nature from the completeness theorems for products of particle-hole operators which are proven to be... 

At this point of our discussion, it may be convenient to take up the problem how one would treat n excitation operators Dx, D2,. . . , Dn constructed from a common basis B of order m in the operator space simultaneously. The excitation operators are given in the form... 

In concluding this subsection, we note that a basis of the ket-bra type P = Pu is always orthonormal according to (2.54), whereas this is usually not true for an arbitrary basis B = Bt, B2,.. . , Bm in the operator space. However, even such a general basis may be brought to orthonormal form by means of successive, symmetric, or canonical orthonormalization.19 Using the symmetric procedure, one obtains, e.g.,... 

It should be observed that it does not matter whether one describes the operator space in terms of particle-hole operators in the language of second quantization or in terms of ket-bra operators. Choosing the ket-bra basis P = Pu constructed from the orthonormal set

carrier space as our basis B, one obtains for r = (k, /) and s = (m, n) ... 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

Let us start by considering a truncated basis B in the operator space which contains m linearly independent elements B = , Bm, ... 

Let us now study the ket-bra operators )(4> formed from the wave functions = B, B < . One knows that, if one goes over to the corresponding orthonormal basis

operator space (which is orthonormal in terms of the Hilbert-Schmidt binary product) is of order pi x p = pc- One has further... 

Both Simons and Yeager employ the 3-block basis operators as the secondary operator space, retaining only portions of the diagonal matrix elements thereof. When the correlation coefficients are calculated by RSPT and the 5-block operators (i.e., a al,a ala and a ala ala ) are Schmidt orthogonalized to the simple electron removal operators (the 1-block), the matrix vanishes through first order. Therefore, the 5-block basis operators do not contribute until fourth order [since (37) is bilinear in. 4 ]. Differences between the approaches of Yeager and Simons are described more fully and tested numerically in Section III.A. 

Fig. 3. Calculated Nj X-ray photoelectron spectrum for incident photons with energy of 1254 eV. The primary operator space for the calculations contains only a minimal number of shake-up-basis operators (repartitioning scheme 1).

Nearly all the peaks in the calculated Nj spectrum have a number of basis operators that contribute significantly. This indicates that the simple molecular orbital picture of the shake-up process is insufficient. " The results emphasize the need for some selection technique, such as the perturbation theory approach employed here, in the choice of the primary operator space. The addition of several extra shake-up basis operators by the perturbation selection criterion lowers the peak from 17.12 to 16.79 eV. The most important of these extra operators involves removal of a electron and de-excitation of a second Itt electron to the lw level. The importance of this operator, which acts only on the correlation part of 0>, is not obvious by pure chemical intuition. 

Fig. 5. EOM and Cl vertical ionization potentials for BH solid line, relaxed Cl long and short dashes, unrelaxed Cl, using SCF orbitals of BH dashed curve, extensive EOM dotted curve, primitive repartitioned EOM. The EOM results are plotted against the tolerance for retaining shake-up-basis operators in the primary operator space, and the dimension of the primary operator space is given in parentheses for each tolerance. The Cl values are presented at the one configuration level (1C), for single and double excitations Cl (SD), and for single, double, and triple excitations Cl (SDT). EOM calculations are not performed at tolerance of 0.01 au because this tolerance does not result in an appreciable increase in the dimensionality of the f -space. Experimental value is 9.77 eV. Asterisk EOM primary operator space restricted to simple ionization operators.

The EOM results for BH are well converged even when the primary operator space is restricted to simple ionization basis operators (the dimension of App is 15), yielding an IP that is 0.2 eV above the best Cl case (SDT, relaxed). Since there are no shake-up basis operators in the P-space, this calculation does not contain any effects due to the 5-block basis opeators or any off-diagonal couplings. (Only the diagonal terms in Aqq through first order are retained.)... 

For example, the observed transitions of an AB spin system have a Liouville matrix given in equation (B2.4.35). The coupling constant is J, and it is assumed that cOg = -co = -5/2, so that 5 is the frequency difference between the two sites. The angle, 0, is defined for the AB system by the equation tan(0)=J/25. The Liouville space basis used here is the superspin equivalent of the four product operators (/, ... 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

Considering the derivation of DKH Hamiltonians so far, we are facing the problem to express all operators in momemtum space, which is somewhat unpleasant for most molecular quantum chemical calculations which employ atom-centered position-space basis functions of the Gaussian type as explained in section 10.3. The origin of the momentum-space presentation of the DKH method is traced back to the square-root operator in Sq of Eq. (12.54). This square root requires the evaluation of the square root of the momentum operator as already discussed in the context of the Klein-Gordon equation in chapter 5. Such a square-root expression can hardly be evaluated in a position-space formulation with linear momentum operators as differential operators. In a momentum-space formulation, however, the momentum operator takes a... 

A scheme [2.15] is presented to follow the evolution of the density matrix in pulse experiments for a system of isolated spin 1 = 1 nuclei. In this case, it is convenient to express a t) in terms of an orthogonal basis set of nine [(2/ + 1) ] 3x3 matrices in an operator space. The choice of this complete basis set is not unique and varies according to the natme of the problem to be solved. A set in which the matrices are Hermitian is chosen so that the spin states represented by these matrices have real physical significance, and the matrices obey convenient conunutation relations with the operators in the Hamiltonian of interest. 

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