Orbital rotations

UNSUBSTITUTED BUTADIENE. Butadiene anchors were presented in Figures 1(3) and 13. The basic tetrahedral character of the conical intersection (as for H4) is expected to be maintained, when considering the re-pairing of four electrons. Flowever, the situation is more complicated (and the photochemistiy much richer), since here p electrons are involved rather than s electrons as in H4. It is therefore necessary to consider the consequences of the p-orbital rotation, en route to a new sigma bond. 

SUBSTITUTED BUTADIENES. The consequences of p-type orbitals rotations, become apparent when substituents are added. Many structural isomers of butadiene can be foiined (Structures VIII-XI), and the electrocylic ring-closure reaction to form cyclobutene can be phase inverting or preserving if the motion is conrotatory or disrotatory, respectively. The four cyclobutene structures XII-XV of cyclobutene may be formed by cyclization. Table I shows the different possibilities for the cyclization of the four isomers VIII-XI. These structmes are shown in Figure 35. 

The orbital rotation is given by a unitary matrix U, which can be written as an exponential transformation. 

One of the goals of Localized Molecular Orbitals (LMO) is to derive MOs which are approximately constant between structurally similar units in different molecules. A set of LMOs may be defined by optimizing the expectation value of an two-electron operator The expectation value depends on the n, parameters in eq. (9.19), i.e. this is again a function optimization problem (Chapter 14). In practice, however, the localization is normally done by performing a series of 2 x 2 orbital rotations, as described in Chapter 13. 

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. 

Carbon-carbon single bonds in alkanes are formed by a overlap of carbon sjy hybrid orbitals. Rotation is possible around a bonds because of their cylindrical... 

Disrotatory (Section 30.2) A term used to indicate that p orbitals rotate in opposite directions during electrocvclic ring-opening or ring closing. 

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

Also the CASSCF/CASPT2 calculations were performed in D2h symmetry and the orbitals rotations were restricted such that mixing between different angular... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to [Pg.474]

Using perturbation dependent atomic orbitals (rotational London orbitals [61]) as basis functions. 

Using perturbation dependent atomic orbitals (rotational London orbitals [61]) as basis functions. Experimental values for the u = 0 state gx =0.5654 + 0.0007 and gn =0.5024 + 0.0005 [74] minus a zero-point-vibrational correction Agx = —0.0135 and Agn = —0.0062 calculated with a... 

The partial sucess/failure slowed down the applications, especially as the computers at that time were too slow to manage larger model spaces and additional, more complicated, A -representability conditions. Some hope was offered by applying symmetries—orbital rotation, spin, isobaric spin—and it was stimulating to explore them with Bob Erdahl and my younger collaborator Bojan Golli [32], However, new ideas were needed. 

Finally, we note that if we retain two-particle operators in the effective Hamiltonian, but restrict A to single-particle form, we recover exactly the orbital rotation formalism of the multiconfigurational self-consistent field. Indeed, this is the way in which we obtain the CASSCF wavefunctions used in this work. 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

So far we have based the formalism on an N-electron basis built from Slater determinants. However, as shown above, both the Hamiltonian and the orbital rotations can be described in terms of the orbital excitation operators Ey. All... 

The first term in this expression is the zeroth order energy E(0,0). The two next terms gives the first derivatives with respect to the parameters Ty (equation (3 37)) and SK0 (equation (3 42)). We obtain the following result for the Erst derivative with respect to the orbital rotation parameters ... 

The solution to this system of equations gives the parameter set T, from which the orbital rotations can be determined (see exercise 3.7). In addition we obtain the variations in the Cl coefficients. Note that we cannot use them directly, since that does not correspond to a unitary rotation, which keeps normalization. Instead we use the defmition S = C SCq to construct the unitary matrix exp(S). The rotated state is then obtained from the formula given in exercise (3.8). 

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

The super-CI method can alternatively be given in a folded form, which includes the coupling between the Cl and orbital rotations. This is done by adding the complementary Cl space, IK>, to the super-CI secular problem. As in the Newton-Raphson approach, it is more efficient to transform the equations back to the original CSF space, and thus work with a super-CI consisting of the Cl basis states plus the SX states. It is left to the reader as an exercise to construct the corresponding secular equation and compare it with the folded one-step Newton-Raphson equations (4 22). 

However, the RAS concept on the other hand complicates the orbital optimization. The wave function is no longer complete in the active orbital space. It is then necessary to introduce orbital rotations between the three RAS spaces, which may lead to convergence problems in more difficult cases. 

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