Nodal Surface Research Articles

Quantum Monte Carlo studies on small molecules

The Variational Monte Carlo (VMC) and Fixed-Node Diffusion Monte Carlo (FNDMC) methods have been examined, through studies on small molecules. New programs have been written which implement the (by now) standard algorithms for VMC and FNDMC. We have employed and investigated throughout our studies the accuracy of the common Slater–Jastrow trial wave function. Firstly, we have studied a range of sizes of the Jastrow correlation function of the Boys–Handy form, obtained using our optimization program with analytical derivatives of the central moments in the local energy. Secondly, we have studied the effects of Slater-type orbitals (STOs) that display the exact cusp behaviour at nuclei. The orbitals make up the all important trial determinant, which determines the fixed nodal surface. We report all-electron calculations for the ground state energies of Li2, Be2, H2O, NH3, CH4 and H2CO, in all cases but one with accuracy in excess of 95%. Finally, we report an investigation of the ground state energies, dissociation energies and ionization potentials of NH and NH+. Recent focus paid in the literature to these species allow for an extensive comparison with other ab initio methods. We obtain accurate properties for the species and reveal a favourable tendency for fixed-node and other systematic errors to cancel. As a result of our accurate predictions, we are able to obtain a value for the heat of formation of NH, which agrees to within less than 1 kcal mol−1 to other ab initio techniques and 0.2 kcal mol−1 of the experimental value.

Characterisation of bicontinuous cubic liquid crystalline systems of phytantriol and water using cryo field emission scanning electron microscopy (cryo FESEM)

Cubosomes are a novel lipid particulate delivery system currently being investigated for drug delivery purposes. The present study investigates bicontinuous cubic liquid crystalline systems (bulk phase and cubosomes) formed by phytantriol and water using cryo field emission scanning electron microscopy (cryo FESEM). Previously cubosomes have been characterized by cryo transmission electron microscopy (cryo TEM) with small angle X-ray diffraction (SAXS) confirming the bicontinuous liquid crystalline type. Bulk cubic phase and cubosomes were prepared from phytantriol and Pluronic F127 and analysed using cryo FESEM and SAXS. The micrographs showed the cubic phase had a tortuous, bicontinuous nature with a non-intersecting network of water channels. The cubosomes also show the same underlying tortuous structure entirely consistent with that of the bulk cubic phase and closely resemble the mathematical description of cubosomes described using nodal surface representation. The structure of both systems was confirmed using SAXS as a bicontinuous cubic liquid crystalline phase with Pn3 m geometry. Cryo FESEM provides valuable insights into the morphological features of bicontinuous cubic liquid crystalline systems. The unique details shown provide strength to support the nodal surface representation of bicontinuous cubic liquid crystalline systems. Cryo FESEM provides a new technique to complement cryo TEM and SAXS for investigating their structure and function.

QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψIwhich hopefully approximates some ground state ψ0of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψIcoincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψIdiffer from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.

Quantum solvation dynamics of HCN in a helium-4 droplet

Ultracold nanodroplets of helium-4, containing several thousands of He atoms, offer considerable promise as microscopic cryogenic chambers. Potential applications include the creation of tailor-made chemical or biomolecular complexes and studies of superfluidity in nanoscale systems. Recent experiments have succeeded in interrogating droplets of quantum solvent which consist of as few as 1-20 helium-4 atoms and which contain a single solute molecule. This allows the transition from a floppy, but essentially molecular, complex to a dissolved molecule to be followed and, surprisingly, the transition is found to occur quite rapidly, in some cases for as few as N = 7-20 solvent atoms. For example, in experiments on helium-4 droplets seeded with CO molecules [Tang and McKellar, J. Chem. Phys. 119, 754 (2003)], two series of transitions are observed which correlate with the a-type (Delta K = 0) and b-type (Delta K = +/-1) lines of the binary complex, CO-He (K is the quantum number associated with the projection of the total angular momentum onto the vector connecting the atom and the molecular center of mass). The a-type series, which evolves from the end-over-end rotational motion of the CO-He binary complex, saturates to the nanodroplet limit for as few as 10-15 helium-4 atoms, i.e., the effective moment of inertia of the molecule converges to its asymptotic (solvated) value quite rapidly. In contrast, the b-type series, which evolves from the free-molecule rotational mode, disappears altogether for N approximately 7 atoms. Similar behavior is observed in recent computational studies of HCN(4He)N droplets [Paolini et al., J. Chem. Phys. 123, 114306 (2005)]. In this article the quantum solvation of HCN in small helium-4 droplets is studied using a new fixed-node diffusion Monte Carlo (DMC) procedure. In this approach a Born-Oppenheimer-type separation of radial and angular motions is introduced as a means of computing nodal surfaces of the many-body wave functions which are required in the fixed-node DMC method. Excited rotational energies are calculated for HCN(4He)N droplets with N = 1-20: the adiabatic node approach also allows concrete physical mechanisms to be proposed for the predicted disappearance of the b-type series as well as the rapid convergence of the a-type series to the nanodroplet limit with increasing N. The behavior of the a-type series is traced directly to the mechanics of angular momentum coupling-and decoupling-between identical bosons and the molecular rotor. For very small values of N there exists significant angular momentum coupling between the molecule and the helium atoms: at N approximately 10 solvation appears to be complete as evidenced by significant decoupling of the molecule and solvent angular momenta. The vanishing of the b-type series is predicted to be a result of increasing He-He repulsion as the number of solvent atoms increases.

Detailed Structure of Diamond-Type Lipid Cubic Nanoparticles

Supramolecular three-dimensional self-assembly of nonlamellar lipids with fragments of the protein immunoglobulin results in a bicontinuous cubic phase fragmented into nanoparticles with open water channels (cubosomes). The structure of the diamond-type cubic nanoparticles is characterized experimentally by freeze-fracture electron microscopy, and it is mathematically modeled with nodal surfaces emphasizing the fluid-like undulations of the cubosomic interfaces. Based on scaling-up and scaling-down approaches, we present stable and intermediate-kind nanoparticles resulting from the cubosomic growth. Our results reveal the smallest stable diamond-type cubosomic entity that can serve as a building block of more complex nanostructured fluid drug delivery vehicles of therapeutic proteins. The evidence presented for lipid-bilayer undulations in the surface region of the protein/lipid cubosomes could have important consequences for possible applications of these hierarchically organized porous nanoparticles.

On the Existence and the Profile of Nodal solutions of Elliptic Equations Involving Critical Growth

We study the existence of sign changing solutions to the slightly subcritical problem $$-\Delta u=|u|^{p-1-\epsilon} u\ {\rm in}\ \Omega,\quad u=0\ {\rm on}\ \partial \Omega,$$ where ω is a smooth bounded domain in ℝ N , N ≥ 3, p = (N + 2)/(N − 2) and ɛ > 0. We prove that, for ɛ small enough, there exist N pairs of solutions which change sign exactly once. Moreover, the nodal surface of these solutions intersects the boundary of ω, provided some suitable conditions are satisfied.

CuAl 2 revisited: Composition, crystal structure, chemical bonding, compressibility and Raman spectroscopy

The structure of CuAl 2 is usually described as a framework of base condensed tetragonal antiprisms [CuAl 8/4]. The appropriate symmetry governed periodic nodal surface (PNS) divides the space of the structure into two labyrinths. All atoms are located in one labyrinth, whereas the second labyrinth seems to be ‘empty’. The bonding of the CuAl 2 structure was analyzed by the electron localization function (ELF), crystal orbital Hamiltonian population (COHP) analysis and Raman spectroscopy. From the ELF representation it is seen, that the ‘empty’ labyrinth is in fact the place of important covalent interactions. ELF, COHP in combination with high-pressure X-ray diffraction and Raman spectroscopy show that the CuAl 2 structure is described best as a network built of interpenetrating graphite-like nets of three-bonded aluminum atoms with the copper atoms inside the tetragonal-antiprismatic cavities.

On factorial double solids with simple double points

We study when double covers of P 3 ramified along nodal surfaces are not Q -factorial. In particular, we describe all the Q -factorial double covers of P 3 ramified along quartic surfaces with at most seven simple double points and sextic surfaces with at most 16 simple double points.

Diffusion Monte Carlo Study of Atomic Systems from Li to Ne

A systematic diffusion Monte Carlo (DMC) study of both neutral and charged atomic systems from Li to Ne in the ground state is performed to evaluate the correlation energy (Ec), the ionization energy (IE), and the electron affinity (EA) of these systems. The present study is based on the fixed-node approximation in which the nodal surfaces of the DMC wavefunction is assumed to be the same as those of the HartreeFock wavefunction. The present calculations reproduce 90 � 7% of the exact value of correlation energy for the cations, 91 � 3% for the neutral atoms, and 92 � 2% for the anions, respectively. The theoretical values of IE and EA in the present study are in good agreement with experimental values within an accuracy of 0.3 eV for IE and 0.1 eV for EA. The variation of Ec and IE with respect to the atomic number Z is interpreted. [doi:10.2320/matertrans.47.2612]

An investigation of nodal structures and the construction of trial wave functions

The factors influencing the quality of the nodal surfaces, namely, the atomic basis set, the single-particle orbitals, and the configurations included in the wave-function expansion, are examined for a few atomic and molecular systems. The following empirical rules are found: the atomic basis set must be fairly large, complete active space and natural orbitals are usually better than Hartree-Fock orbitals, multiconfiguration expansions perform better than single-determinant wave functions, but only few configurations are effective and their choice is suggested by symmetry considerations, while too long determinantal expansions spoil the nodal surfaces. These rules allow us to reduce the nodal error and to compute the best fixed node-diffusion Monte Carlo energies for a series of dimers of first-row atoms.

Flux expansion nodal method for solving multigroup neutron diffusion equations in hexagonal- z geometry

A flux expansion nodal method (FENM) has been developed to solve multigroup neutron diffusion equations in hexagonal- z geometry. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group. In order to improve the nodal coupling relations, a new type of nodal boundary conditions is proposed, which requires the continuity of both the zero- and first-order moments of partial currents across the nodal surfaces. The response matrix technique is used for the iterative solution of the nodal diffusion equations, which greatly improves the computational efficiency. The numerical results for a series of benchmark problems show that FENM is a very accurate and efficient method for the prediction of criticality and nodal power distributions in the reactors with hexagonal assemblies.

A complex ball uniformization of the moduli space of cubic surfaces via periods ofK3 surfaces

In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K 3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for different covers of the moduli space corresponding to geometric markings of the Picard group or a choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.

Classification of atomic states by geometrical and quantum-mechanical symmetries

The eigenstates of nonrelativistic two-, three-, four-, and six-valence-electron atoms are classified according to their transformation properties under symmetry operations. In energy-favorable geometrical arrangements of the electrons, the spatial distribution of the wave function is constrained by the demand of invariance under rotation, inversion and permutation of particles. For a given term, the symmetry constraints may leave the energy-favorable arrangement accessible or inaccessible. In the latter case the term is not expected to belong to the low-energy part of the spectrum. Contrary, in the former case with no symmetry-induced inherent nodal surface in the multidimensional coordinate space of the multielectron wave function, the term is expected to belong to the low-energy part of the spectrum. The symmetry-based classification is confirmed by recent calculations.

All-electron quantum Monte Carlo calculations for the noble gas atoms He to Xe

We report all-electron variational and diffusion quantum Monte Carlo (VMC and DMC) calculations for the noble gas atoms He, Ne, Ar, Kr, and Xe. The calculations were performed using Slater-Jastrow wave functions with Hartree-Fock single-particle orbitals. The quality of both the optimized Jastrow factors and the nodal surfaces of the wave functions declines with increasing atomic number Z, but the DMC calculations are tractable and well behaved in all cases. We discuss the scaling of the computational cost of DMC calculations with Z.

Influence of molecular symmetry on strong-field ionization: Studies on ethylene, benzene, fluorobenzene, and chlorofluorobenzene

Using the molecular strong-field approximation we consider the effects of molecular symmetry on the ionization of molecules by a strong, linearly polarized laser pulse. Electron angular distributions and total ionization yields are calculated as a function of the relative orientation between the molecule and the laser polarization. Our studies focus on ethylene (C$_2$H$_4$), benzene (C$_6$H$_6$), fluorobenzene (C$_6$H$_5$F), and ortho chlorofluorobenzene (1,2 C$_6$H$_4$ClF), the molecules representing four different point groups. The results are compared with experiments, when available, and with the molecular tunneling theory appropriately extended to non-linear polyatomic molecules. Our investigations show that the orientational dependence of ionization yields is primarily determined by the nodal surface structure of the molecular orbitals.

Hyperspherical analysis of radial correlations in four-electron atoms

Hyperspherical coordinates are used to study electron correlations in the radial motion of a four-electron atom within the model of s{sup 4} configurations. We identify groups of hyperspherical adiabatic potential curves that support singly, doubly, triply, and quadruply excited states. By examining the nodal structures of the channel functions, we show that it is possible to relate the number of nodal surfaces in the hyperangles and the principal quantum numbers used in the independent particle picture.

Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball

In [41], it was shown that the following singularly perturbed Dirichlet problem e 2 Δu - u + |u| p- u = 0, in Ω, u = 0 on ∂Ω has a nodal solution u e which has the least energy among all nodal solutions. Moreover, it is shown that u e has exactly one local maximum point P e 1 with a positive value and one local minimum point P e 2 with a negative value, and as e → 0, formula math where φ(P 1 , P 2 ) = min(}P 1 - P 2 }/2,d(P 1 , ∂Ω ), d(P 2 , ∂Ω)). The following question naturally arises: where is the nodal surface {u e (x) = 0}? In this paper, we give an answer in the case of the unit ball Ω = B 1 (0). In particular, we show that for e sufficiently small, Pf, P e 2 and the origin must lie on a line. Without loss of generality, we may assume that this line is the x 1 -axis. Then u e must be even in Xj, j = 2,..., N, and odd in x 1 . As a consequence, we show that {u e (x) = 0} = {x ∈ B 1 (0) x 1 = 0}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P e 1 , P e 2 and the origin must lie on the x 1 -axis and u e must be even in Xj, j = 2,..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u e (which implies the odd symmetry of u e in x 1 ). Similar results are also proved for the problem with Neumann boundary conditions.

On factoriality of nodal threefolds

We prove the Q-factoriality of a nodal hypersurface in P4 of degree n with at most (n−1) 2 4 nodes and the Q-factoriality of a double cover of P3 branched over a nodal surface of degree 2r with at most (2r−1)r 3 nodes.

A general approach to the calculation of2H2O NMR lineshapes in microheterogeneous systems: a distorted bicontinuous cubic phase

A new computational method is developed for calculating 2H NMR lineshapes of H2O in microheterogeneous systems, such as lyotropic liquid crystals that exhibit curved lipid/water interfaces. The method presented is based on the stochastic Liouville equation (SLE) in its Langevin form. This means that the Liouville equation of motion is combined with Brownian dynamics simulations to describe the stochastic spin-lattice Liouvillian. The NMR relaxation is caused by translational diffusion of the heavy water molecules, along the curved 2H2O/lipid interface. The model used is a nodal surface approximation of the cubic symmetric gyroid minimal surface. This unit cell is then isotropically expanded or distorted in two dimension. The changes in 2H20 NMR lineshapes have been calculated for the enlarged or the distorted cubic unit cell. The timescale of the residual quadrupole interaction, which determines the NMR lineshape, ranges from the Redfield regime to the slow-motional regime depending on the curvature of the interface. The distortion of the cubic phase illustrates the possibility to explore the intermediate interfaces of a phase transition, by means of 2H2O lineshape analysis.

The molecular physics lecture 2004: (i) Density functional theory, (ii) Quantum Monte Carlo

In this combination of review and new science, we shall look at Density Functional Theory (DFT) and the Quantum Monte Carlo (QMC) method. For DFT, we shall (i) discuss the inadequacy of the Uniform Electron Gas model for chemistry (ii) examine Generalized Gradient Approximation (GGA) exchange functionals (iii) understand that they introduce left–right correlation for molecules (iv) argue that the functionals BLYP and OLYP are parameter-free for molecules (v) look at the ‘remarkable’ performance of these local functionals in the prediction of binding energies (vii) discuss hybrid functionals, and show how they can be improved through the introduction of an Ewald partitioning of 1/r 12, already examined by Hirao, Savin and Gill (viii) show that our new functional CAM-B3LYP offers good energetics as well as good charge transfer excitation energies, already shown by Hirao. For QMC, concentrating on the Diffusion Monte Carlo approach, we shall (i) relate the trial wavefunction to the transcorrelated wavefunction, and look again at the transcorrelated method, (ii) look at the nodal surfaces of the trial wavefunction, demonstrating in particular that the use of finite basis sets in the Hartree–Fock method necessarily introduces ‘spurious’ nodal features which probably cannot be ignored, (iii) argue that, now it is possible to perform Hartree–Fock calculations with slater-type basis sets, it is straightforwardly possible to obtain orbitals which obey the exact nuclear cusp conditions, (iv) therefore argue that these basis sets should be used in Monte Carlo studies.