Free‐energy differences between states with different conformational ensembles

Multiple conformations separated by high‐energy barriers represent a challenging problem in free‐energy calculations due to the difficulties in achieving adequate sampling. We present an application of thermodynamic integration (TI) in conjunction with the local elevation umbrella sampling (LE/US) method to improve convergence in alchemical free‐energy calculations. TI‐LE/US was applied to the guanosine triphosphate (GTP) to 8‐Br‐GTP perturbation, molecules that present high‐energy barriers between the anti and syn states and that have inverted preferences for those states. The convergence and reliability of TI‐LE/US was assessed by comparing with previous results using the enhanced‐sampling one‐step perturbation (OSP) method. A linear interpolation of the end‐state biasing potentials was sufficient to dramatically improve sampling along the chosen reaction coordinate. Conformational free‐energy differences were also computed for the syn and anti states and compared to experimental and theoretical results. Additionally, a coupled OSP with LE/US was carried out, allowing the calculation of conformational and alchemical free energies of GTP and 8‐substituted GTP analogs. © 2013 Wiley Periodicals, Inc.     

Reparameterization of hybrid functionals based on energy differences of states of different multiplicity

 Low-spin/high-spin energy splittings for Fe(II) transition-metal complexes – particularly in weak ligand fields – cannot be well described by density functional methods. Different density functionals yield results which differ by up to 1 eV in transition-metal complexes with sulfur-rich first coordination spheres. We attribute this failure to the fact that the high-spin state is systematically favoured in Hartree–Fock-type theories, because Fermi correlation is included in the exact exchange, while Coulomb correlation is not. We thus expect that the admixture of exact exchange to a given density functional will heavily influence the energy splitting between states of different multiplicity. We demonstrate that the energy splitting depends linearly on the coefficient of exact exchange admixture. This remarkable result is found for all the Fe(II)–S complexes studied. From this observation we conclude in connection with experimental results that Becke's 20% admixture should be reduced to about 15% if meaningful energetics are sought for transition-metal compounds. We rationalize that this reduction by 5% will not affect the quality of the hybrid functional since we arrive at a slightly modified functional, which lies between the pure density functional and the hybrid density functional, which both give good results for “standard” systems. Received: 13 July 2001 / Accepted: 31 August 2001 /  Published online: 16 November 2001     

Hall's variational principle for excited states

It is shown how excited state energies can be given upper bounds through the use of Hall's functional. Applications to atomic central field problems are worked out.     

A structure registry condition for the interaction energy between two solids

The interaction energy between two periodic crystals obeys a structure registry condition. When the 2D periodic structures in the atomic planes parallel to the surfaces of the two crystals are not coincident, the energy contribution which accounts for the discrete characteristics of the solids along these planes vanishes. The remaining continuum potential is then unable to discriminate between the atomic distribution of surface atoms when one crystal is laterally translated with respect to the other. This registry condition leads to drastic changes in the interaction energy between two ionic crystals and can be responsible for the disappearance of specific structural properties in solids.     

An approximate relation between Wigner-Seitz-type one-electron Hamiltonians and complete Hamiltonians for molecules and solids

Summary The relation between the sum of Wigner-Seitz-type one-electron Hamiltonians of all electrons in a system (any molecule or solid) and the complete Hamiltonian for the system is deduced. According to this relation, the total energy for an electronic configuration may be formulated approximately as a quadratic function of the atomic charges defined for the configuration added to the sum of the energies of occupied orbitals. This formula is necessary for evaluating the system's total energy, using the orbital energies calculated from the Wigner-Seitz-type potential (called the linkage of embedded atomic fields (LEAF)), and is also useful for analyzing and understanding electronic structures, reactivities, and other properties of molecules and solids.     

A variational principle for the excitation energy of a multielectron system

A variational principle is proposed for direct calculation of the excitation energy. The spectrum of collective excitations is derived for a homogeneous electron gas.     

Constraints in the variational principle for stationary states

Constraints in the variational principle for stationary states (VPSS) are classified in accordance with Dirac’s constrained classical mechanics and the time-dependent variational principle (TDVP). All of the VPSS constraints are required to belong to the first-class TDVP as constants of motion to ensure the real-valuedness of the Lagrange multipliers. The VPSS constraints are further classified as either first-class or second-class. The first-class VPSS constraints are constants of variation with symmetry-adapted wave functions. If the representation basis for the constraint operators is incomplete, however, the first-class VPSS constraints lead to broken-symmetry solutions. The nondegenerate energies of \({}^2E'\) at the \(D_{3h}\) geometry in the Jahn–Teller distortion of H\(_3\) are presented as an example of a problem with broken-symmetry. An ad hoc adjustment is suggested by considering the second-class pseudo-VPSS constraints with new adiabatic parameters.     

Coexistence of two different anion states in polyacene nanocluster anions

Two types of anion states are shown to coexist in nanometer-scale polyacene cluster anions. Naphthalene and anthracene nanoclusters having a single excess electron were produced in the gas-phase. Photoelectron spectra of size-selected cluster anions containing 2 to 100 molecules revealed that rigid "crystal-like" cluster anions emerge, greater than approximately 2 nanometers in size, and coexist with the "disordered" cluster anion in which the surrounding neutral molecules are reorganizing around the charge core. These two anion states appear to be correlated to negative polaronic states formed in the corresponding crystals.     

A field theoretical description of states with different orbitals for different spins

A field theoretical formulation is given for the method of different orbitals for different spins (DODS ). For an infinite system DODS describes an antiferromagnetic state and to account for this spin ordering a Gorkov-type of factorization is introduced. A corresponding gap equation is derived, where a non-zero solution indicates the presence of long-range order in the system. Actually the formulation given is general enough as to render DODS as a special case. As shown, we may also obtain a ferrimagnetic as well as a density wave state solution depending on the special characteristics of the system at hand. A related type of antiferromagnetism is described by the Overhauser spin density wave (SDW ) state and also this theory is formulated in a field theoretical language. The similarities and differences between DODS and SDW are discussed. The energy expressions for the two states are given within the Hartree-Fock approximation. It is proposed that the SDW state could be used to partly account for the correlation problem in molecules, as well as the method of DODS which has previously been employed for that purpose.     

A variational principle for weak stimulated absorptions with energy minimizing fields

Optimal time-dependent perturbations in agreement with the Bohr condition for spectroscopic transitions are obtained variationally for two-level systems via the first-order perturbation model. The criterion of optimality is that specified weak absorptions are achieved with minimal applied energy. Results obtain for fixed perturbation intervals of arbitrary length.     

Time-dependent variational principle for nonlinear Hamiltonians and its application to molecules in the liquid phase

The formulation of the time-dependent Frenkel variational principle for Hamiltonians containing a term depending on the wave function is here considered. Starting from the basic principles, it is shown that it requires the introduction of a related functional, G, which, for the systems we are considering, has the status of a free energy. An explicit use of functional G as starting point to obtain variational wave functions makes it easier to implement computational methods for a variety of physical and chemical problems in solution, the first one among them being the calculation of frequency-dependent nonlinear optical properties of components of the liquid phase. A concise overview of applications of this approach which are presently being worked out in our laboratory is also given. © 1996 John Wiley & Sons, Inc.     

A reduced-restricted-quasi-Newton–Raphson method for locating and optimizing energy crossing points between two potential energy surfaces

We present a method for the location and optimization of an intersection energy point between two potential energy surfaces. The procedure directly optimizes the excited state energy using a quasi-Newton–Raphson method coupled with a restricted step algorithm. A linear transformation is also used for the solution of the quasi-Newton–Raphson equations. The efficiency of the algorithm is analyzed and demonstrated in some examples. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :992–1003, 1997     

A corresponding states principle for the equation of state of hard body fluid mixtures

A theoretically based corresponding-states principle is developed for athermal mixtures consisting of hard molecules. The principle states that when scaled appropriately, the excess compressibility factor for such mixtures reduces to a universal function of the effective packing fraction of the mixture. The latter represents the number density reduced by means of the effective molecular volume, which is defined as the volume a molecule excludes to any point of another molecule and depends on the geometry of both molecules. The scaling factor is related to a sort of effective nonsphericity parameter for the mixture that depends on composition as well as the nonsphericity parameters of the molecules which form the mixture and their effective molecular volumes. The universal function represents the excess compressibility factor of a pure hard-sphere fluid. Results are in good agreement with available simulation data.