Water dimer vibration-rotation tunnelling levels from vibrationally averaged monomer wavefunctions

The vibration-rotation tunnelling (VRT) spectra for the water dimer obtained by vibrationally averaging the dimer potential over accurate water monomer wavefunctions is reported. The vibrational averaging requires evaluation of the 12D dimer potential energy surface at more than 10¹² distinct geometries. The resulting vibrational spectra of the low-lying dimer states are presented and compared with both less computationally expensive methods based on fixed nuclei approximations and the recent (6+6)d adiabatic calculations of Leforestier et al. [38] (2009). The procedure gives some modest improvement in the agreement with experimental values for the vibration-rotation tunnelling (VRT) states of (H₂O)₂ and (D₂O)₂. This approach can be extended to treat dimer states involving monomer overtone excitations, which is important in obtaining water dimer spectra at infrared and visible wavelengths at atmospheric temperatures, and in characterizing the dimer contribution to the so-called water continuum absorption at these wavelengths.     

On the improvement of expectation values in the uncoupled SCF and HF perturbation theories

The exact derivation of the formula for the first-order ‘correlation’ correction to the second-order uncoupled Hartree-Fock energies, computed with approximate Hartree-Fock orbitals, is presented. A comparison with the corresponding formula derived under the assumption of the exact solution of the Hartree-Fock equations indicates that also the second-order perturbed orbitals do contribute. However, their contribution can be expressed in terms of the first-order correction to the SCF orbitals which accounts for the difference between the Hartree-Fock and SCF solution. These additional terms can be either positive or negative and can influence significantly the final result for the correlation-corrected second-order energies.     

Instantons and vacuum expectation values in the Schwinger model

Vacuum expectation values for the Schwinger model are computed within the path integral formalism for an instanton vacuum. The calculations are done in flat Euclidean space and use the wellknown anomaly result. It can be demonstrated that the instanton vacuum is absolutely necessary to obtain a consistent quantization. At last the results are compared to a similar treatment on the sphere and to the operator solution of the Schwinger model.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Projekt Nr. P8444-TEC     

Conditional expectation values in quantum mechanics

The general question of defining the expectation value of an operator for a fixed value of another noncommuting observable is considered and explicit expressions are derived. Due to the noncommutivity of operators a unique definition is not possible, and we consider different possible expressions. Special cases which have previously been considered in the literature are shown to be derivable from the methods presented.In honor of Ilya Prigogine on the occasion of his 70th birthday.     

Far-infrared VRT spectroscopy of two water trimer isotopomers: vibrationally averaged structures and rearrangement dynamics

We report the measurement of far-infrared vibration-rotation tunnelling parallel bands of two partially deuterated water trimer isotopomers: (D2O)2DOH and (H2O)2DOH at 97.2607 cm-1 and approximately 86 cm-1, respectively. The hydrogen bond rearrangement dynamics of the two mixed trimers can be described by the simplified molecular symmetry G8, which accounts for both the flipping and bifurcation tunnelling motions previously established for (H2O)3 and (D2O)3. The observed donor tunnelling quartet, rather than triplet, splitting indicates that the two homogeneous monomers (D2O or H2O) in each mixed trimer experience slightly different environments. Vibrationally averaged structures of (H2O)3, (D2O)3, and (D2O)2DOH were examined in a Monte Carlo simulation of the out-of-plane flipping motions of the free atoms. The simulation addresses both the symmetric top behaviour and the negative zero-point inertial defect for (H2O)3 and (D2O)3, which were insufficiently counted in all previous structure models. The average ground state O--O separations, which are correlated to other angular coordinates, were determined to be 2.84 +/- 0.01 angstroms for all three species. The simulated difference in hydrogen bond nonlinearity also supports the inequivalency of the two homogeneous monomers. The structural simulation shows that the unique H in (D2O)2DOH is free, while a torsional analysis suggests the unique D in (H2O)2DOH is bound within the cyclic ring. Both bands can be assigned to the pseudorotational transitions which correlate to those found in the pure trimers.     

Finite temperature expectation values of boundary operators

A conjecture is presented for the thermal one-point function of boundary operators in integrable boundary quantum field theories in terms of form factors. It is expected to have applications in studying boundary critical phenomena and boundary flows, which are relevant in the context of condensed matter and string theory. The conjectured formula is verified by a low-temperature expansion developed using finite size techniques, which can also be used to evaluate higher point functions both in the bulk and on the boundary.     

Asymptotic estimates of expectation values for atom

The asymptotic estimates of the operators 〈r^a〉 and 〈p^b〉 are presented for all physically permissible values of exponents both for a neutral atom and for an atom without electron-electron interaction. The improved estimate of the operator ?2〉 including two exact terms is obtained.     

Nonperturbative flow equations from running expectation values

We show that Wegner's flow equations, as recently discussed in the Lipkin model, can be solved self-consistently. This leads to a nonlinear differential equation which fully determines the order parameter as a function of the dimensionless coupling constant, even across the phase transition. Since we consider an expansion in the fluctuations, rather than the conventional expansion in the coupling constant, convergence to the exact results is found in both phases when taking the thermodynamic limit.     

Vacuum expectation values from fusion of vertex operators

The algebra of fused vertex operators for the ABF model is defined and studied in the free fields approach. Vacuum expectation values of local operators in the scaling theory are reproduced from the matrix elements of the fused vertex operators.     

Area expectation values in quantum area Regge calculus

Regge calculus generalised to independent area tensor variables is considered. Continuous time limit is found and formal Feynman path integral measure corresponding to the canonical quantisation is written out. Quantum measure in the completely discrete theory is found which possesses the property to lead to the Feynman path integral in the continuous time limit whatever coordinate is chosen as the time. This measure can be well defined by passing to the integration over imaginary field variables (area tensors). Averaging with the help of this measure gives finite expectation values for areas.     

Vacuum expectation values of the energy-momentum tensor in two dimensions

A general form of the vacuum expectation values of the components of the energy-momentum tensor is derived in the case of static two-dimensional spacetime. Conditions for the regularity of the energy-momentum tensor are set. A generalized formula for the black hole temperature originally found by Hawking is given. The regularity of the energy-momentum tensor in the presence of more than one horizon is investigated.On leave of absence from the Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland.     

Fluctuations of expectation values in chaotic systems and broken time-reversal symmetry

Fluctuations of expectation values of observables are calculated in complex quantum systems, such as disordered metallic grains or quantum systems with classical chaotic motion. We derive an exact expression for these fluctuations valid for systems with and without time-reversal symmetry, as well as in the transition region between these two cases. We compare our results with those of a semiclassical theory and with simulations of random matrices.     

A new hierarchy of upper and lower bounds on expectation values

Upper and lower bounds are constructe for expectation values of functions of a real random variable with derivatives up to orderN+1 which are alternately negative and positive over the whole range of interest. The bounds are given by quadrature formulas with weights and abscissas determined by the firstN+1 moments of the underlying probability distribution. Application to a simple disordered phonon system yields sharp bounds on the specific heat.