Introduction to linear and nonlinear programming

This paper contains an investigation of spaces with a two parameter Abelian isometry group in which the Hamilton-Jacobi equation for the geodesies is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. The spaces satisfying the stronger condition that the corresponding Schrodinger equation is separable are isolated in a canonical form for which Einstein’s vacuum equations and the source-free Einstein-Maxwell equations (with or without a Λ term) can be solved explicitly. A fairly extensive family of new solutions is obtained including the previously known solutions of de Sitter, Kasner, Taub-NUT, and Kerr as special cases.

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Hamilton-Jacobi and Schrödinger separable solutions of Einstein's equations

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

https://iopscience.iop.org/article/10.1088/0034-4885/35/1/306/pdf

Semiclassical approximations in wave mechanics

The complete active space (CAS) SCF method is presented in detail with special emphasis on computational aspects. The CASSCF wave function is formed from a complete distribution of a number of active electrons in a set of active orbitals, which in general constitute a subset of the total occupied space. In contrast to other MCSCF schemes, a CASSCF calculation involves no selection of individual configurations, and the wave function therefore typically consists of a large number of terms. The largest case treated here includes 10 416 spin and space adapted configurations. To be able to treat such large CI expansions, a density‐matrix oriented formalism is used. The Newton–Raphson scheme is applied to calculate the orbital rotations, and the secular problem is solved with recent developments of CI techniques. The applicability of the method is demonstrated in calculations on the HNO molecule in ground and excited states, using a triple‐zeta basis and different sizes of the active space. With a reasonable choice of active space, the calculations converge in 6–10 iterations. This is true also for states which are not the lowest state of the symmetry in question. The equilibrium geometry for the ground state is RNO=1.215(1.212) A, RNH =1.079(1.063) A, ϑHNO=108.8(108.6) °, the experimental values given in parenthesis for comparison. The best estimates for the transition energies to the lowest 3A″ and 1A″ states are 0.67(0.85) eV and 1.52(1.63) eV, respectively. The results obtained indicate that the choice of active space may be crucial for the convergence properties of CASSCF calculations.

The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule

The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.

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Classical equations for quantum systems

A variational method is presented which is applicable to N‐particle boson or fermion systems with two‐body interactions. For these systems the energy may be expressed in terms of the two‐particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground‐state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N‐particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two‐particle and one‐particle density matrices of an N‐particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ...

Reduction of the N‐Particle Variational Problem

A path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.

Hamiltonian Path-Integral Methods

A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = JHijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.

The variational approach to the two−body density matrix

By making the assumption that the addition (subtraction) of an electron from a many-body wave function may be represented by a single spin orbital, variationally determined, two distinct one-particle potentials are derived. They represent, respectively, the effective interaction of the (N - 1)-electron system with the Nth electron, and the effective interaction of the N-electron system with the (N + 1)th electron. Nonlocal exchange arises because of the indistinguishability of electrons. The “core” electrons are allowed to be correlated with each other and to be polarized by the electron in question. Eigenvalues to these potentials represent, respectively, ionization potentials and electron affinities, and the eigenfunctions are variationally determined spin orbitals (“standing-wave solutions”), termed the “natural transition orbitals.”



The two-body density matrix of a reference state approximating an eigenfunction of the many-body Hamiltonian is required as input to the theory. Given this input, the method is linear, requiring no interations. In the limit of no correlation in the reference N-body state (provided the occupied orbitals span the Hartree-Fock space), the two potentials reduce to the unrestricted Hartree-Fock potential for occupied and virtual orbitals.

A generalization of the hartree‐fock one‐particle potential

A detailed study is made of the properties of the ``particle‐hole matrix'' Gabcd≡〈g| (a†b−ρab)†(c†d−ρcd) |g〉, where |g> is a many‐fermion ground state and ρab is the 1‐body density matrix. It is shown that the zero eigenvalues of the particle‐hole matrix are intimately and simply related to the one‐body symmetries of the ground state. It is also shown that the large eigenvalues of Gabcd are closely related to the collective features of the ground state.

Claude Garrod

Papers

Reduction of the N‐Particle Variational Problem

Hamiltonian Path-Integral Methods

The variational approach to the two−body density matrix

A generalization of the hartree‐fock one‐particle potential

Particle--hole matrix: its connection with the symmetries and collective features of the ground state.