Commutator Expansion Research Articles

Unitary coupled-cluster based self-consistent polarization propagator theory: A quadratic unitary coupled-cluster singles and doubles scheme.

The development of a quadratic unitary coupled-cluster singles and doubles (qUCCSD) based self-consistent polarization propagator method is reported. We present a simple strategy for truncating the commutator expansion of the unitary version of coupled-cluster transformed Hamiltonian H̄. The qUCCSDmethod for the electronic ground state includes up to double commutators for the amplitude equations and up to cubic commutators for the energy expression. The qUCCSDexcited-state eigenvalue equations include up to double commutators for the singles-singles block of H̄, single commutators for the singles-doubles and doubles-singles blocks, and the bare Hamiltonian for the doubles-doubles block. Benchmark qUCCSDcalculations of the ground-state properties and excitation energies for representative molecules demonstrate significant improvement of the accuracy and robustness over the previous UCC3 scheme derived using Møller-Plesset perturbation theory.

Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

On the propagation of regularity for solutions of the dispersion generalized Benjamin–Ono equation

In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and Benjamin-Ono equation, satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In this case the nonlocal term present in the dispersive generalized Benjamin-Ono equation is more challenging that the one in BO equation. To deal with this a new approach is needed. The new ingredient is to combine commutator expansions into the weighted energy estimate. This allow us to obtain the property of propagation and explicitly the smoothing effect.

Towards Near-Exact Solutions of Molecular Electronic Structure: Full Coupled-Cluster Reduction with a Second-Order Perturbative Correction.

We introduce a new augmented adaptation of the recently developed full coupled-cluster reduction (FCCR) with a second-order perturbative correction, abbreviated as FCCR(2). FCCR is a selected coupled-cluster expansion aimed at optimally reducing the excitation manifold and commutator expansions for high-rank excitations for obtaining accurate solutions of the electronic Schödinger equation in a size-extensive manner. The present FCCR(2) enables estimating the residual correlation of FCCR by the second-order perturbative correction E(2) from the complementary space of the FCCR projection manifold. The linear relationship between E(2) and the energy of FCCR(2) allows accurate estimates of near-exact energies for a wide variety of molecules with strong electron correlation. The potential of the method is demonstrated using challenging cases, the ground-state electronic energy of the benzene molecule in equilibrium and stretched geometries, and the isomerization energy of the transition metal complex [Cu(NH3)]2O22+.

Operator Splitting Around Euler-Maruyama Scheme and High Order Discretization of Heat Kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Commutator expansions in NMR diffusometry

In this paper commutator expansions for solving the Bloch–Torrey’s equation are derived. An exact solution for free diffusion in a constant magnetic field gradient is found. Furthermore the moments of the signal in the short gradient pulse limit for spins confined to a restricted geometry are derived. The moments are tested numerically against the known solution of diffusing spins between two perpendicular plates in the short gradient pulse limit.

On higher order estimates in quantum electrodynamics

We propose a new method to derive certain higher order estimates in quantum electrodynamics. Our method is particularly convenient in the application to the non-local semi-relativistic models of quantum electrodynamics as it avoids the use of iterated commutator expansions. We re-derive higher order estimates obtained earlier by Fröhlich, Griesemer, and Schlein and prove new estimates for a non-local molecular no-pair operator.

Cluster expansions of canonically transformed hamiltonians and effective interactions

The Van Kampen cluster expansion method is adapted to the problem of resuming the commutator expansion arising in a canonical transformation upon the Hamiltonian operator of a many-electron system. The canonical transformation generates effective interactions which in the cluster formalism may be grouped in a natural way according to the number of interacting particles. A practical scheme is developed for exhibiting the many-body terms with explicit formulas given for the two-body effective interactions. The formalism is applied to the σ-π separability problem for planar molecules, with the generator of the transformation chosen to eliminate, to a certain approximation, the σ-π interactions. This procedure yields an effective π-π interaction which is exhibited explicitly as an expansion over the spectrum of two-particle a states. The results closely resemble those of Harris though the procedure arising from the cluster expansion approach is somewhat more straightforward to carry out.

Lie algebras and Lie groups over noncommutative rings

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra ( g , A ) ( F ) , which is the Lie subalgebra of F ⊗ A generated by F ⊗ g . In many examples A is the universal enveloping algebra of g . Our description of the algebra ( g , A ) ( F ) has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. To each algebra ( g , A ) ( F ) we associate a “noncommutative algebraic” group which naturally acts on ( g , A ) ( F ) by conjugations and conclude the paper with some examples of such groups.

One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory

One-electron density matrices resulting from the explicitly connected commutator expansion of the expectation value were implemented at the singles and doubles coupled cluster (CCSD) level. In the proposed approach the one-electron density matrix is obtained at a little extra cost in comparison to the calculation of the CCSD correlation energy. Therefore, in terms of the computational time the new method is significantly less demanding than the conventional linear-response CCSD theory which requires additionally an expensive calculation of the left-hand solution of the CCSD equations. The quality of the new density matrices was investigated by computing a set of one-electron properties for a series of molecules of varying sizes and comparing the results with data obtained using the full configuration interaction method or higher level coupled cluster theory. It has been found that the results obtained using the new approach are of the same quality as those predicted by the linear-response CCSD method. The novel one-electron density matrices have also been applied to study the energy of the electrostatic interaction for a number of van der Waals complexes, including the benzene and azulene dimers.

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

In [M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73–118] a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit description of this quantization is given in Theorem 2.5. A description of the A-module structure of the Poisson envelope of a smooth commutative algebra A was given in loc. cit., Theorem 4.1.3. However the proof given in loc. cit. has a gap. We fix this gap for A local (Theorem 1.4) and prove a weaker global result (Theorem 1.6).

De Rham and Infinitesimal Cohomology in Kapranov's Model for Non commutative Algebraic Geometry

The title refers to the nilcommutative or NC-schemes introduced by M. Kapranov in ‘Noncommutative Geometry Based on Commutator Expansions’, J. Reine Angew. Math 505 (1998) 73–118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson or NP-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC- and NP-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposes sur la cohomologie des schemas, Masson, Paris (1968), pp. 306–358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.

TIME-DEPENDENT SCATTERING THEORY OF N-BODY QUANTUM SYSTEMS

We give a full and self contained account of the basic results in N-body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r-μ, [Formula: see text]. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t → ±∞ for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions.

Direct renormalization of two-electron self-energy contribution to energy of helium-like ions

A new practical approach to the calculation of two-electron self-energy corrections is presented for the energy of helium-like ions. The working formulas are derived within the modified adiabatic formalism. The direct renormalization procedure is used in terms of multiple commutator expansions. It is shown that the final expression does not contain ultraviolet and infrared divergences. This expression is used then to obtain numerical estimates for two-electron self-energy corrections in wide range of the nuclear charge Z.

Calculated self-energy contributions for annsvalence electron using the multiple-commutator method

The self-energy (SE) correction is evaluated for a single valence $\mathrm{ns}$ electron of heavy and superheavy atoms with n up to 8 and the nuclear charge Z up to 119. The recently developed approach based on the commutator expansion is employed. Various Dirac-Slater one-electron local potentials with extended nuclei are used. The Lamb shifts were calculated by adding the average values of the Uehling potential to the SE contributions. The results confirm the earlier estimates for the quantum electrodynamical effects on the valence energies of heavy and superheavy atoms.

A new Approach to the Electron Self-Energy Calculations

A new method for the calculation of the first-order self-energy for tightly bound electrons is presented. The method is based on the multiple commutator expansion and the partial-wave renormalization.

Noncommutative geometry based on commutator expansions

We develop an approach to noncommutative algebraic geometry ``in the perturbative regime around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.

Noncommutative geometry based on commutator expansions

We develop an approach to noncommutative algebraic geometry ``in the perturbative regime around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.

Varieties of Anticommutativen-ary Algebras

A fundamental problem in the theory ofn-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describes some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteristic 0 any variety ofn-ary algebras can be defined by multilinear identities. In the anticommutative case, it is shown that forn≤8 the[formula]-dimensionalS2n−1-module of multilinear identities in which each term involves twon-ary products (i.e., two pairs ofn-ary anticommutative brackets) decomposes as the direct sum of thendistinct simple modules labelled by thenpartitions of 2n−1 in which only 1 and 2 occur as parts. In the casesn=3 (resp.n=4), the kernel of the commutator expansion map and a generator for each of the 7 (resp. 15) nonzero submodules are determined. The paper concludes with some conjectures forn≥5.

Multiple commutator expansion for the Lamb shift in a strong Coulomb field

A new expression for the electron self-energy in a strong Coulomb field is obtained without expansion in powers of ( is the fine structure constant, Z is the charge of the nucleus). This expression contains the multiple commutator expansion, which differs from the expansion and converges rapidly even for very high Z values. The renormalization procedure requires the proof of a new sum rule. This proof is given explicitly for Z = 1 and implicitly for all Z values for the state. The numerical calculation of the Lamb shift for different Z values is performed and comparison with the known results made.