J. Čížek

On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell‐Type Expansion Using Quantum‐Field Theoretical Methods

A method is suggested for the calculation of the matrix elements of the logarithm of an operator which gives the exact wavefunction when operating on the wavefunction in the one‐electron approximation. The method is based on the use of the creation and annihilation operators, hole—particle formalism, Wicks theorem, and the technique of Feynman‐like diagrams. The connection of this method with the configuration‐interaction method as well as with the perturbation theory in the quantum‐field theoretical form is discussed. The method is applied to the simple models of nitrogen and benzene molecules. The results are compared with those obtained with the configuration‐interaction method considering all possible configurations within the chosen basis of one‐electron functions.

Time-Independent Diagrammatic Approach to Perturbation Theory of Fermion Systems

Publisher Summary The chapter discusses the time-independent diagrammatic approach to perturbation theory of fermion systems. The chapter explores the perturbation theory for a non-degenerate level. The formulas derived serves as a starting point for the subsequent consideration of the excitation and ionization energies. The advantages of the direct calculation of excitation energies, compared with the approach in which the total energies of the pertinent electronic states are calculated separately for each state and then the excitation energies are obtained by subtracting the appropriate state energies, are quite obvious. The Rayleigh-Schrodinger (RS) perturbation theory (PT) for the case of a non-degenerate level of some Hamiltonian operator is discussed. The chapter discusses that even the Rayleigh-Schrodinger perturbation expressions for the direct calculation of the excitation energies may be obtained in a rather simple way without the involvement of the Green function formalism. On the contrary, our simple approach using the ordinary perturbation theory for separate levels presents certain desirable features of the Green function formalism. The chapter explains the diagrammatic representation of Wicks theorem and resulting diagrams. General explicit formulas for the second- and third-order excitation energy contributions are given in the chapter.

Coupled cluster approach or quadratic configuration interaction

It is shown that a recently proposed quadratic configuration interaction (QCI) method, when limited to single and double substitutions (QCISD), represents a special case of the single reference coupled cluster approach. When applied to higher levels of substitutions (QCISDT) the method ceases to be size extensive. The relationship of QCISD method with existing coupled cluster approaches is shown in detail.

Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. IV. A Study of Doublet Stability for Odd Linear Polyenic Radicals

The doublet stability conditions for the restricted Hartree–Fock (HF) solutions of the simple open‐shell case (i.e., one electron in addition to the closed shell) are applied to the π‐electronic models of the odd linear polyenic radicals, using the Pariser–Parr–Pople‐type semiempirical Hamiltonian. It is found that symmetry‐adapted restricted HF solutions for linear polyenes with (4ν − 1) carbon atoms are always doublet stable while those with (4ν − 1) carbon atoms may be unstable for large enough coupling constants (i.e., the ratio of two‐electronic part of the Hamiltonian to the one‐electronic part). In fact the HF solutions for the latter systems are generally doublet unstable already for the currently used values of the semiempirical parameters or a very small lowering of the absolute value of the resonance integral β (i.e., ∼0.1 − 0.3 eV) is sufficient to yield instability for any (4ν − 1) type polyene. The instability, in fact, occurs already for the allyl radical, which is studied in a considerable detail. The doublet stability of the solutions for the (4ν + 1) type polyenes is explained on the basis of symmetry properties of corresponding Kekule structures. The symmetry‐nonadapted restricted HF solutions are calculated for illustration of their properties and possible multiplicities.

Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. III. Rules for the Singlet Stability of Hartree–Fock Solutions of π‐Electronic Systems

The singlet stability conditions for closed‐shell electronic systems, which ensure that the Hartree–Fock (HF) determinant with doubly occupied orbitals minimizes the energy expectation value, are applied to the symmetry adapted HF solutions of linear polyacenes, using the Pariser–Parr–Pople‐type semi‐empirical Hamiltonian. It is found that the symmetry adapted HF solutions for linear polyacenes containing an even number of benzene rings are always singlet stable, while the HF solutions for linear polyacenes having an odd number of benzene rings may exhibit singlet instability if the coupling constant is large enough. For cases where singlet instability was found, we have also calculated new HF solutions having lower energy than the symmetry adapted HF solutions. These new HF solutions violate the space symmetry conservation laws as usual. Furthermore, the qualitative rules for the existence of singlet stability of the symmetry adapted HF solution of π‐electronic systems with conjugated double bonds are de...

Alternancy symmetry: A unified viewpoint

A general formulation of the alternancy symmetry adaptation for the semiempirical Pariser–Parr–Pople (PPP) type Hamiltonians is presented at both the spin‐orbital and spin‐adapted many‐electron levels. The derivation of the general form of the alternancy symmetry conjugation operators is based solely on the tight‐binding approximation for the short range one‐particle part of the Hamiltonian considered. It starts by a simple formulation of the desired invariance properties of the PPP type Hamiltonian. Using algebraic properties of the unitary group generators and of their particle number nonconserving extensions, it leads to a completely explicit and general form for the alternancy symmetry conjugation operators. In this way the prior descriptions, which become special cases of this general formulation, are interrelated and unified. The spin and quasispin character of certain components of these operators are also pointed out and explicitly derived. The spin‐adapted version is based on the unitary group fo...

Lie Algebraic Methods and Their Applications to Simple Quantum Systems

Publisher Summary The algebraic methods were first introduced in the context of the new matrix mechanics around 1925. The importance of the concept of angular momentum in quantum mechanics, in contrast to classical mechanics was soon recognized and the necessary formalism was developed principally by Wigner, Weyl, and Racah. The algebraic treatment of the angular momentum can be found nowdays in almost every textbook on quantum mechanics, often in parallel with the differential equation approach. In contrast, this cannot be said about another basic quantum mechanical system, namely the hydrogen atom, which is almost universally treated by the latter approach only. In the midst of various attempts to solve this difficult problem, the elementary particle physicists examined several noncompact Lie algebras in the mid-l950s, hoping that these physicists can provide a clue to the classification of elementary particles. A close relationship of the angular momentum and the so(3) algebra dates certainly to the prequantum mechanics era, while the realization that SO(4) is the symmetry group of the Kepler problem was first demonstrated by Fock. Soon afterward Bargmann showed that the generators of Focks SO(4) transformations were the angular momentum and Runge–Lenz vectors.

Numerical application of the coupled cluster theory with localized orbitals to polymers. IV. Band structure corrections in model systems and polyacetylene

We present the formalism for the correction of the band structure for correlation effects of polymers in the framework of a localized orbital approximation, using the quasiparticle model. For this purpose we use in an ab initio framework Mo/ller–Plesset perturbation theory in second order, the coupled cluster doubles method, and its linear approximation. The formalism is applied to a water stack and two different forms of a water chain as model systems to test the reliability of the approximations involved. From our previous work we know that, e.g., in polyacetylene difficulties due to the localizability of the canonical crystal orbitals do not arise from the π or π* bands, but from bands of σ symmetry. Thus we concentrate in this work again on polyacetylene as an example of a realistic polymer. We find that the localized orbital approximation is quite useful also in the case of band structure corrections due to correlation effects. However, the coupled cluster calculations, in particular, turn out to be ...

Very accurate summation for the infinite coupling limit of the perturbation series expansions of anharmonic oscillators

Abstract A new method for the summation of the renormalized perturbation expansions for the quartic, sextic, and octic anharmonic oscillators is proposed. Our method is based on a special nonlinear sequence transformation. Summation results for the infinite coupling limit of the ground state energies are presented and compared with the results obtained by other techniques.

rational approximations for the modified Bessel function of the second kind

Various different rational approximations for the modified Bessel function Kv(z) are compared with respect to their ability of computing Kv(z) efficiently and reliably in the troublesome region of moderately large arguments z. The starting point for the construction of the rational approximations is the asymptotic series 2F0 for Kv(z), which diverges for all finite arguments z but is Borel summable and Stieltjes summable. The numerical tests showed that Pade approximants for Kv(z) are significantly less efficient than the other rational approximations which were considered. The best results were produced by some recently derived sequence transformations (E.J. Weniger, Comput. Phys. Rep. 10 (1989) 189), which are closely related to Levins sequence transformations (D. Levin, Int. J. Comput. Math. B 3) (1973) 371).