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Cite this chapter as: Yajnik U. (1997) Quantum Field Theory Methods: Dirac Equation and Perturbation Theory. In: Iyer B.R., Vishveshwara C.V. (eds) Geometry, Fields and Cosmology. Fundamental Theories of Physics (An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application), vol 88.

Classical matrix perturbation bounds, such as Weyl (for eigenvalues) and David-Kahan (for eigenvectors) have, for a long time, been playing an important role in various areas: numerical analysis. Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. Theory of cosmological perturbations (PR215, 1992)(T)(131s).djvu │ Roos M. M(atrix) theory. Matrix quantum mechanics as fundamental theory (RMP73, 2001)(43s).pdf. Win32,Mac CDROM ISO (movies, Mma notebooks).rar. Matrix Perturbation Theory and its. Yaniv Shmueli. School of Computer Science. Tel-Aviv University. November 29, 2012. Yaniv Shmueli (TAU).

Springer, Dordrecht. DOI. Publisher Name Springer, Dordrecht. Print ISBN 978-90-481-4902-5. Online ISBN 978-94-017-1695-6. eBook Packages.

164 Spin–OrbitCoupling in Molecules energies E ð k 1;a Þ are obtained by diagonalizing the interaction matrix in a basis spanned by the d components j ð k 0;b Þ i of j ð k 0 Þ i, where b is an index from 1 to d. 0 h ð 0Þ ^ ð 0Þ h ð 0Þ ^ ð 0Þ h ð 0Þ ^ ð 0Þ 1 ð 0;Þ j ^ j ð 0; Þ i ð 0; Þ j ^ j ð 0;Þ i. Ð 0;Þ j ^ j ð 0;Þ i k 1 HSO k 1 k 2 HSO k 1 k d HSO k 1 C B h k; 1j H. SOj k; 2i h k; 2j H. SOj k; 2i.

SOj k; 2i 190 B. C @ A B 0 H^ SO 0 Þ 0 Þ H^ SO 0. 0 H^ SO 0 C B ð Þ ð h ð ð Þ h ð Þ ð Þ C B h k; 1 j j k;d i k; 2 j j k;d i k;d j j k;d i C Only one of the matrix elements needs to be evaluated explicitly. All others can be obtained from the reduced matrix element by means of the Wigner– Eckart theorem, Eq.

The eigenvectors j k ð 0;a Þi ¼ d jck ð 0;a Þ;b j k ð 0;b Þ i ½191 & X b¼ 1 are zeroth-orderwave functions adapted to this particular perturbation. Computational Aspects 165 channel. Near the equilibrium geometry, the low-lyingelectronic states are usually well separated. Their mutual interaction is then described properly by second-orderperturbation theory. In the separated atom limit, first-orderdegenerate perturbation theory applies. In between, but still close to the dissociation limit, the states are nearly degenerate—or quasi-degenerateas we might say. Neither of the procedures is then strictly applicable and orders of perturbation theory are not well defined.

Quasi-DegeneratePerturbation Theory An often chosen way out of this dilemma is to set up a so-calledperturbation matrix in the basis of eigenvectors of a spin-freesecular equation, multiplied by an appropriate spin function. In this basis, matrix elements of the spin-freeHamiltonian occur only in the diagonal, of course. Due to symmetry, diagonal spin–orbitmatrix elements come along only with complex wave functions. In a Cartesian basis, the integrals of the electrostatic Hamiltonian and the y component of ^ are real, whereas x and z integrals exhibit an HSO imaginary phase. (The spatial parts of the integrals are imaginary for all Cartesian components. It is the choice of an imaginary phase for S ^ y that makes the matrix elements of the y component of ^ real.) As before, matrix elements HSO are computed only for one representative of a spin multiplet; all other matrix elements are generated by use of the WET.

Similar to first-orderdegenerate perturbation theory, perturbed energies and wave functions are obtained by matrix diagonalization. This approach has therefore been named quasi-degenerateperturbation theory. In spite of this designation, the procedure is also applicable to states with large energy separations.

Other authors prefer to call it LS contracted spin–orbitconfiguration interaction (SOCI) in order to stress its relation to configuration interaction procedures in the limit of a complete set of eigenvectors. 108 The perturbation matrix is in most cases small enough for the application of standard complex Hermitian eigenvalue and eigenvector solvers. Otherwise one can resort to the same methods as in SOCI. One of the great advantages of quasi-degenerateand Rayleigh–Schro¨dinger perturbation theories over SOCI procedures is that different levels of sophistication can easily be mixed. For reasons of efficiency or technical limitations, it is in general not possible to a perform a full CI calculation that yields the exact solution within a given basis set. Instead, configurations are selected according to some criterion such as excitation class or energy. Unfortunately, electron correlation contributions are slowly convergent.

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Furthermore, truncated CI expansions are not size extensive, that is, the correlation energy does not scale properly with the number of electrons. For spin-freestates, long-standingexperience exists on how to estimate correlation contributions from discarded configurations or excitation classes. 109–115 These extrapolated energies or the eigenvalues of correspondingly dressed Hamiltonians can be taken as diagonal elements combined with spin–orbitmatrix. 166 Spin–OrbitCoupling in Molecules elements of single excitation CI or smaller single and double excitation CI expansions. 41,108,116 A weakness of these methods lies in the limited number of zeroth-orderstates that are used for an expansion of the first-orderperturbed wave function.

In particular, it has been demonstrated that probabilities of spin-forbid-den radiative transitions converge slowly with the length of the perturbation expansion. 92 Variational Perturbation and Response Theory As an alternative to sum-over-statesmethods, the perturbation equations can be solved directly. In the context of spin–orbitcoupling, reviews on this subject have recently been given by Yarkony 117 ˚ 118 and by Agren et al.

119 utilized a Hamiltonian matrix approach to determine the spin–orbitcoupling between a spin-freecorrelated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony 120 proposed to solve the matrix equation Hð 0Þ Eð 0Þ 1 Þ ð 1Þ ¼ H ð 0 Þ ½ 194 & ð i i SO i directly for ð i 1 Þ in the basis of CSFs. The direct solution of such a perturbation equation is generally known as Hylleraas (or variational) perturbation theory.

121,122 Yarkony 123 developed this approach further by introducing a sym- bolic matrix element method, thereby extending the limits relating to the dimensionality of the Hamiltonian matrix. Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin–orbitmatrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbiddenradiative transition probabilities. We refrain from going into details here, because an ˚ 118 While excellent review on this subject has been published by Agren et al. These authors focus on response theory and its application in the framework of CI and multiconfiguration self-consistentfield (MCSCF) procedures, an analogous scheme using coupled-clusterelectronic structure methods was presented lately by Christiansen et al. 124 Variational Procedures For compounds containing heavy atoms, spin–orbitand electron correlation energies are approximately of the same size, and one cannot expect these effects to be independent of each other.

A variational approach that treats both interactions at the same level is then preferable to a perturbation expansion. Special care is advisable in the choice of the spin–orbitoperator in this case. The variational determination of spin–orbitcoupling requires a spin–orbit. 168 Spin–OrbitCoupling in Molecules formulation, the Davidson method is applicable to real symmetric matrices only. With some care, it can nevertheless be applied also to a complex eigenvalue problem. Every complex Hermitian matrix C can be written as C ¼ A þ iB ½195 & where A and B are real matrices. Since C is Hermitian, A is a symmetric matrix, and B is skew symmetric.

G Similarly, the eigenvectors Z i of C can be itself is real. Divided into a real part U i and an imaginary part iV i, where V i The complex eigenvalue problem ½196 & CZ i ¼ E i Z i can then be replaced by a real symmetric eigenvalue problem of two times its original dimension. U i U i B A V i V i ½197 & All eigenvalues E i of the real eigenvalue problem 197 are doubly degenerate. This degeneracy is of purely technical origin and should not be confused with Kramers degeneracy, 139,140 which may occur in addition. (For instance, four degenerate roots are obtained for a doublet state, i.e., two for each Kramers t level.) In addition to the transposed eigenvector ðU i V i Þ, a second one with t structure ð V i U i Þ is obtained. One of these solutions can be discarded. Alter- natively, root-homingprocedures can be employed to avoid its evaluation right from the beginning. Lets playsix of the best games for mac.

136,137 Older versions of SOCI programs are very I/O intensive because they used to store the Hamiltonian matrix on disk and read it in every iteration step. 52,141 Integral-drivendirect methods for spin–orbitcoupling came up in the mid 1980s 123,142 following the original fomulation of direct CI methods for spin-independent Hamiltonians.

143–146 Modern direct SOCI programs can easily handle several million determinants. 108,147–151 The convergence of the iterative determination of eigenvalues and eigenvectors is accelerated appreciably if spin–orbitCI and quasi-degenerateperturbation theory procedures are combined. To this end, the perturbation matrix is set up in the basis of the most important LS contracted CI vectors j ð m 0; 1 Þ i. The solutions of this small eigenvalue problem j k ð 1; 1 Þi ¼ sf CI states X c km ð 1; 1 Þj m ð 0; 1 Þi ½198 & m are then used to start up the Davidson iteration.

Computational Aspects 169 As the number of electrons increases, the dimension of the CI space becomes increasingly large, even if excitations are restricted to single and double replacements. Many approximate schemes to restrict the number of configurations in a SOCI have been devised. The selected intermediate coupling CI (SICCI) code is an extension of the original COLUMBUS DGCI program and utilizes electrostatic and magnetic interactions as a selection criterion. 152 The SPDIAG program 141 within the BNSOC package 153 is based on the MRD-CI approach. The latter makes use of a correlation energy criterion for configuration selection and estimates the contribution of the discarded configurations to the spin-freecorrelated energy E ð m 0; 1 Þ by means of Epstein–Nesbetperturbation theory. 111 If one assumes that the expansion 198 represents a decent approx- imation to the SOCI solution, the MRD-CIextrapolation scheme can easily be extended to the spin–orbitcoupled case.

154,155 Also other approaches toward a balanced treatment of spin–orbitinteraction and electron correlation are based on a manipulation of the spin-freeenergies and wave functions. A pure shift of excitation energies in the Davidson 135 start-upmatrix is not suf- ^ ficient, because the original eigenvalues of H ð 0 Þ are restored during the iterative process. 141 In the so-called spin-freestate shifted (SFSS) SOCI method, this problem is circumvented by introducing a projector on the set of ð m 0; 1 Þ. 156 sf CI Xstates ^ sfss ^ E m j ð 0; 1Þ ih ð 0; 1Þ j ½ 199 & HSO ¼ H SO þ m m m with E m ¼ ð E m E XÞ ð Eð 0; 1Þ Eð 0; 1Þ Þ ½ 200 & m X The E m quantities shift the spin-freeexcitation energies ðE ð m 0; 1 Þ E ð X 0; 1 Þ Þ, calculated at a lower level of correlation treatment, to the exact values ðE m E X Þ or at least to higher accuracy estimates.

Herein, X denotes a common reference state, in general the electronic ground state. Due to the structure of the spin–orbitHamiltonian, off-diagonalblocks of the CI matrix are dominated by single excitations. On the other hand, for electron correlation effects (diagonal blocks) double and higher excitations are decisive. In many approximate schemes, this fact is exploited. One of the first of these approaches was the relativistic CI (RCI) algorithm by Balasubramanian. 157 This is a two-stepprocedure. In a first step, large-scalemulticonfiguration SCF and multireference singles and doubles CI (MRSDCI) calculations are carried out for a spin-freeHamiltonian to determine a set of natural orbitals (NOs).

In the second step, spin–orbitinteraction integrals are transformed to the NO basis and added to the one-electronHamiltonian matrix elements of a smaller, truncated MRSDCI in relativistic symmetries. DiLabio and Christiansen 158 studied the separability of spin–orbitand correlation energies for.

170 Spin–OrbitCoupling in Molecules sixth-rowmain group compounds. They propose to determine the energy shift D SO s due to spin–orbitcoupling as the difference between spin-freeand intermediate coupling single excitation CI calculations. The energy of the correlated spin–orbitcoupled states is then estimated by summing D SO s into the spin-freecorrelated energies E ls sd. Rakowitz et al. 159 employed a SFSS SOCI approach on the Ir þ ion. They demonstrated that the heavily spin–orbitperturbed spectrum of this ion can be obtained in good accord with experiment at the single excitation level, if higher level correlated electrostatic energies are used to determine the shifts E m. Recently Vallet et al.

160 extended the effective Hamiltonian approach, originally implemented in the CIPSO code at the level of quasi-degenerateperturbation theory, 41 to be used in intermediate coupling eff ^ calculations. Instead of representing H el þ H SO in the basis of LS contracted states, the new effective and polarized spin–orbitCI (EPCISO) operates in a determinantal model space. This model space is chosen as the union of the separate reference spaces in nonrelativistic symmetries augmented by the most significant configurations contributing to either the correlation energy or spin–orbitcoupling. COMPARISON OF FINE-STRUCTURESPLITTINGS WITH EXPERIMENT Spectroscopic parameters of a molecule are derived from experimentally determined spectra by fitting term values to a properly chosen model Hamiltonian. 161 Usually, the model Hamiltonian is an effective one-stateHamiltonian that incorporates the interactions with other electronic states parametrically.

In rare cases, experimentalists have used a multistate ansatz like the supermultiplet approach 162 to fit the rovibronic spectra of strongly interacting near-degenerateelectronic states. The safest way of comparing theoretical data to experiment is to compute the spectrum and to fit the calculated term energies to the same model Hamiltonian as the experimentalists use. The vibrational dependence of an effective molecular parameter A is usually expressed as X ak vi þ d k A v i ¼ A e þ i ½201 & k 2 where A e is the value of the property at the equilibrium distance, and d i is the degeneracy of the vibrational mode v i. The vibrational dependence is caused by the anharmonicity of the potential energy surface and by the variation of A with the vibrational coordinate Q. Such information can in practice be obtained from a quantum chemical treatment only for diatomic and triatomic molecules. To this end, A is fitted to a function of Q—mostlya polynomial. Comparison of Fine-StructureSplittings with Experiment 171 a cubic spline—andvibrationally averaged, that is, AðQÞ is weighted with the probability density of a particular vibrational state w v i ðQÞ and integrated over the vibrational coordinate Q A v i ¼ h w v i ðQÞjAðQÞj w v i ðQÞi ½202 & Alternatively, expectation values computed at the equilibrium geometry of a given electronic state can be directly compared with experimental parameters extrapolated from rovibrational branches.

In many instances, the ‘‘fine-structure’’splitting caused by (first-order) spin–orbitcoupling is considerably larger than the energy separation between adjacent rotational levels—atleast for low values of the total angular momentum J—androtational excitations can thus be neglected in a first approximation. Hydrides are exceptional in this respect because of their low reduced masses and the resulting large rotational constants B. Methods for computing rovibrational spectra of diatomic molecules from ab initio data including spin– orbit and rotational coupling were proposed among others by Yarkony 117 and by the author 163 employing Hund’s case ðbÞ and ðaÞ basis functions, respectively. 161 For a review on the theoretical determination of rovibronic spectra of triatomic molecules, please refer to the work of Peric´ et al.

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164 A general expression for the rotational dependence of a spectroscopic parameter cannot be given. Its functional form varies with the type of basis functions chosen for describing the actual rotation of the nuclear frame; the choice of basis functions in turn depends on the order in which the angular momenta are coupled (Hund’s cases in linear molecules) and the type of rotor spherical top, symmetric top (prolate, oblate), asymmetric top. Looking at these particular cases in detail goes far beyond the scope of the present chapter. For additional reading, see, for example, Refs. First-Order Spin–OrbitSplitting Only spatially degenerate states exhibit a first-order zero-fieldsplitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin–orbitHamiltonian have to be equated with those of a phenomenological operator.

One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-ordercontributions. Second-orderSOC may be large, particularly in heavy element compounds.

As discussed in the next section, it is not always distinguishable from first-ordereffects. A phenomenological spin–orbitHamiltonian, formulated in terms of tensor operators, was presented already in the subsection on tensor operators. Few experimentalists utilize an effective Hamiltonian of this form (see Eq.

Instead, shift operators are used to represent space and spin angular. 172 Spin–OrbitCoupling in Molecules Ω = 3/2 ASO Figure 14 Spin–orbitsplitting pattern of a Ω = 1/2 (regular) 2 r state. (The dashed line marks the position of the unperturbed state.) ASO 0 momentum operators in their tensorial forms.

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Employing shift operator conventions and making use of the Russell–Saunders( LS) coupling scheme, the phenomenological spin–orbitHamiltonian reads ^ ^ 1 ^ ^ ^ ^ ^ ^ A SO L S ¼ a k L 0S 0 þ 2 a?ð L þS þ L S þÞ ½203 & For the sign of the spin–orbitparameter A SO, the following conventions apply:. A positive A SO denotes a regular state (Figure 14). In regular states the sublevel with smallest J or quantum number is lowest in energy. A negative A SO denotes an inverted state (Figure 15). In inverted states the sublevel with largest J or quantum number is lowest in energy. Typically, states with less than half-filledshells (particle states) are regular, whereas states with more than half-filledshells (hole states) are inverted.

Atoms: The Lande´ Interval Rule To determine the first-ordersplitting pattern of an atomic state in terms of the phenomenological spin–orbitparameter A SO Eq. 203, we utilize Ω = 1 2 A SO Ω = 2 2 A SO Figure 15 Spin–orbitsplitting pattern of an Ω = 3 ASO. Comparison of Fine-StructureSplittings with Experiment 173 Russell–Saunderscoupling, that is2 2 ^ ^ ^ ^ ^ ^ ¼ 2 ^ J ¼ ð L þ S Þ. From J ¼ ð L þ S Þ ^ ^ ^ L þ 2L S þ S, we obtain 1 2 2 2 ^ ^ ^ ^ ^ ½204 & L S ¼ 2 ð J L S Þ Exploiting the fact that in first-orderperturbation theory all sublevels exhibit identical ^ 2 and S ^ 2 eigenvalues, respectively, yields a spin–orbitsplitting of L EðJÞ EðJ 1Þ ¼ A SOJ ½205 & for two neighboring atomic fine-structurelevels with quantum numbers J and J 1.

This expression is the famous Lande´ interval rule. As an example where this rule can be applied favorably to determine an atomic spin–orbitparameter, consider the first excited state of atomic copper, 2 D g ðd 9 s 2 Þ. The 2 D g state of copper is well separated from other electronic states that are allowed by symmetry to couple in second order. A 2 D state gives rise to two fine-structurelevels with total angular momentum quantum numbers J ¼ 3 2 and J ¼ 5 2. The d 9 s 2 configuration corresponds to a hole state, and we therefore expect an inverted splitting pattern (Figure 16). From Lande´’s interval rule, we determine a splitting of 5 2 A SO. A SO may thus be calculated directly from the experimentally determined splitting 168 of 2042 cm 1 by multiplying with 2 5, yielding A SO ¼ 816:8 cm 1; this value is in good agreement with the theoretically determined spin–orbitparameter of A SO ¼ 802:4 cm 1.

169 Things are not as easy for the 3 D g ðd 9 s 1 Þ ground state of nickel. From a 3 D atomic state, three fine-structurelevels originate with total angular momentum quantum numbers J ¼ 1, J ¼ 2, and J ¼ 3 (Figure 17). The ratio of the measured fine-structuresplittings ½Eð2 Þ Eð1 Þ&=½Eð3 Þ Eð2 Þ& amounts to 1.234 instead of 2 3 as expected from the Lande´ interval rule. The main cause for this deviation is an energy shift of the J ¼ 2 level due to the interaction with neighboring states of the same angular momentum quantum number. The J ¼ 3 and J ¼ 1 levels of 3 D g are only weakly perturbed and their energy separation, EðJ ¼ 3 Þ EðJ ¼ 1 Þ ¼ 5 A SO, may be taken to extract an experimental value for A SO ¼ 301:6 cm 1.

The experimental spin–orbit constant is in excellent agreement with the theoretical value of 301 cm 1. J = 3/2 5/2 A SO Figure 16 Spin–orbitsplitting pattern of an J = 5/2 inverted 2 D atomic state.