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Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena

Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena

by Frank H. Stillinger

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This book presents an authoritative and in-depth treatment of potential energy landscape theory, a powerful analytical approach to describing the atomic and molecular interactions in condensed-matter phenomena. Drawing on the latest developments in the computational modeling of many-body systems, Frank Stillinger applies this approach to a diverse range of


This book presents an authoritative and in-depth treatment of potential energy landscape theory, a powerful analytical approach to describing the atomic and molecular interactions in condensed-matter phenomena. Drawing on the latest developments in the computational modeling of many-body systems, Frank Stillinger applies this approach to a diverse range of substances and systems, including crystals, liquids, glasses and other amorphous solids, polymers, and solvent-suspended biomolecules.

Stillinger focuses on the topography of the multidimensional potential energy hypersurface created when a large number of atoms or molecules simultaneously interact with one another. He explains how the complex landscape topography separates uniquely into individual "basins," each containing a local potential energy minimum or "inherent structure," and he shows how to identify interbasin transition states—saddle points—that reside in shared basin boundaries. Stillinger describes how inherent structures and their basins can be classified and enumerated by depth, curvatures, and other attributes, and how those enumerations lead logically from vastly complicated multidimensional landscapes to properties observed in the real three-dimensional world.

Essential for practitioners and students across a variety of fields, the book illustrates how this approach applies equally to systems whose nuclear motions are intrinsically quantum mechanical or classical, and provides novel strategies for numerical simulation computations directed toward diverse condensed-matter systems.

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Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena

By Frank H. Stillinger


Copyright © 2016 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-7397-5


Potential Energy Functions

Potential energy functions that control nuclear motions in various forms of matter provide a central object of theoretical attention in the formalism to be developed in subsequent chapters. The quantum mechanics of electrons interacting with the nuclei provides the source of those potential energy functions. It is natural, then, to begin this volume that is devoted to multidimensional potential energy landscapes and their implications by reviewing the relevant underlying quantum mechanical fundamentals. Understanding these fundamentals is a necessary prerequisite for selecting many-body models that are typically used for analytical theory, for computer simulation, and for interpreting experimental observations of a wide range of material systems, both in and out of thermal equilibrium. This chapter includes a survey of some of the commonly encountered model potentials and discusses their principal characteristics for representing both "simple" and "complex" substances.

A complete review of quantum mechanical basics would include relativistic effects that become important when heavy elements with high nuclear charges are present. However, the development to follow is limited to the nonrelativistic regime that is described by the Schrödinger equation. For present purposes, this is a justifiable simplification because it is quantitatively accurate for many applications, and even when relativistic effects exist, this nonrelativistic approximation can still qualitatively capture the majority of properties needed for the development of the energy landscape/inherent structure formalism.

A. Quantum Mechanical Basis

The natural starting point for consideration of condensed-matter properties is the quantum mechanics of the constituent electrons and nuclei, regarded as point particles with fixed masses and electrostatic charges. The dynamical evolution of collections of these point particles is described by the time-dependent Schrödinger equation [Pauling and Wilson, 1935; Schiff, 1968]:

HΨ = -([??]/i)([partial derivative]Ψ/[partial derivative]t) (I.1)

This basic partial differential equation is to be solved subject to applicable initial and boundary conditions, where the system volume in general may be either finite or infinite. Here the Hermitian operator H is the spin-independent Hamiltonian operator for the collection of electrons and nuclei, Ψ is the time-dependent wave function for that collection, and [??] is Planck's constant h divided by 2π. In addition to time t, Ψ also has as its variables the set of electronic and nuclear coordinates, to be denoted respectively by {x'i} [equivalent to] {r'i, s'i} and {xj} [equivalent to] {rj, sj}, where these coordinates include both spatial positions (r', r), and spins if any (s', s). Because of the linearity of the Schrödinger equation, Ψ can be resolved into a linear combination of components corresponding to each of the eigenfunctions of the time-independent wave equation.


In these expressions, n indexes the time-independent spin and space eigenfunctions ψn with respective energy eigenvalues En. The constants An are determined by the initial state of the system under consideration. Because H is spin independent, each ψn can also be chosen simultaneously as an eigenfunction of total spin-squared (S2) and directionally projected spin operators (SZ) for the electrons and each nuclear species present. The ψn are obliged to exhibit the exchange symmetries required for the particle types that are present (symmetric for identical bosons, antisymmetric for identical fermions).

A basic simplification upon which the formalism to be developed in this volume rests is the separation of electronic and nuclear degrees of freedom in the full wave mechanics, using the Born-Oppenheimer approximation [Born and Oppenheimer, 1927]. This approximation is motivated and justified by the wide discrepancy between the mass me of the light electrons on one side and the masses Mj of the much heavier nuclei on the other. In the hypothetical limit for which all ratios me/Mj of electron to nuclear masses approach zero, the eigenfunction and eigenvalue errors committed by the Born-Oppenheimer approximation would also asymptotically approach zero [Takahashi and Takatsuka, 2006]. In most applications to be considered in this text, the actual errors attributable to the Born-Oppenheimer approximation with nonzero mass ratios are in fact negligibly small.

To apply the Born-Oppenheimer approximation, the full Hamiltonian operator H is first separated into two parts,

H = HI + HII (I.3)

where HI refers to motion of electrons in the presence of fixed nuclei, and HII collects all remaining terms (nuclear kinetic energy, and nuclear pair Coulomb interactions). Because the full Schrodinger Hamiltonian H is spin-independent, and if no external forces are present, the specific forms are and




As indicated earlier, the r'i in these equations denotes spatial positions of the electrons, and the rj does the same for spatial positions of the nuclei. Following normal convention, e represents the fundamental Coulomb charge of a proton, and the nuclei bear respective charges Zje. This separation permits the full quantum mechanical problem to be resolved into an ordered sequence of two simpler problems. In the first stage, eigenfunctions and eigenvalues are obtained just for the operator HI, i.e., for the electrons moving in the static Coulomb field supplied by the nuclei at fixed positions, with those electronic eigenfunctions subject to the necessary antisymmetry conditions [Pauling and Wilson, 1935; Schiff, 1968]:


As the vertical-bar notation for the wavefunctions emphasizes, the entire set of electronic eigen-functions and eigenvalues depends parametrically on the nuclear coordinates. For each electron-subsystem indexing quantum number n' there exists a well-defined limit for the eigenvalue as all nuclei recede from one another to infinity:


This limiting energy consists of a sum of independent contributions from the individual atoms and/or ions that are formed as the electrons become partitioned and localized around the widely separated nuclei.

The second stage of the solution sequence addresses the nuclear quantum mechanics. This involves the operator HII augmented by the relevant position-dependent energy eigenvalue from the electron subsystem. Consequently, the effective nuclear Hamiltonian becomes


Here Φ({rj}|n') is the Born-Oppenheimer potential energy (hyper)surface on which the nuclei move:


Notice that En'(el)(∞) has been inserted into Φ to provide a convenient energy origin for the nuclear subsystem. In other words, this convention implies that Φ = 0 when all nuclei are infinitely far from one another. Nuclear eigenfunctions and eigenvalues (indexed by n") are subsequently to be determined by solving the nuclear Schrödinger equation:


subject to given boundary conditions, and to symmetry requirements imposed by the presence of identical nuclei with identical spin components (symmetric under exchange for bosons, anti-symmetric under exchange for fermions) [Pauling and Wilson, 1935, Chapter XIV; Landau and Lifshitz, 1958a, Chapter IX].

As a consequence of this sequential solution process, the Born-Oppenheimer approximations to the full-system eigenfunctions and eigenvalues have the following forms:



The index ("quantum number") n for the initial problem has been replaced by the pair n', n" for electronic and nuclear problems, respectively. It is possible and usually convenient to take the electronic wave functions to be real and orthonormal among themselves for any nuclear configuration. Because of the presence of unbound electrons in highly excited states, this method requires that the system be at least temporarily confined to a finite volume V, which can be allowed to pass to infinity at a later stage. Consequently one can write


The same can be assumed true for the nuclear eigenfunctions for any given electronic quantum state n':


The "integrations" indicated in these last two equations implicitly include spin summations where necessary. The notation δ(j, k) stands for the Kronecker delta function (equal to unity for j = k, zero otherwise).

In parallel with Eq. (I.2), the time dependence of the nuclear quantum dynamics on the n' electronic potential surface can then be expressed as follows:


with numerical coefficients Bn" determined by initial conditions. In circumstances where very heavy nuclei are involved, moving through regions of the configuration space where potential Φ varies slowly with position, quantum dynamics often can be replaced by its classical limit. The corresponding Newtonian equations of motion for the N nuclei then describe the nuclear configurational dynamics [Goldstein, 1953]:


One might legitimately question whether the "bare" nuclear masses Mj that appear in Eqs. (I.8) and (I.16) are appropriate or whether a set of effective masses M*j that include contributions from the electron subsystem would lead to a more precise description. If the electrons are in their lowest energy (ground) state, or in a low-lying excited state, each nucleus would tend to have bound to it a number of electrons equal, or nearly equal, to its atomic number Zj. Because these bound electrons would preferentially move with that nucleus, their mass should in principle be added to the bare nuclear mass. Thus, if the physical system of interest corresponds to well-separated electrostatically neutral atoms (e.g., noble gas atoms), the effective masses would be

M*j = Mj + Zjme, (1.17)

where me is the electron mass. If the nuclei exist within the system as ions with fixed oxidation states (electrostatic charges), then expression (I.17) would have to be modified accordingly to account for the deficit or surfeit of electrons compared to the atomic number Zj. Chemical bonding to produce molecular species or extended covalent networks within which included nuclei execute vibrational motions present a more complicated situation; detailed calculations would be necessary to reveal the extent to which electron mass follows the vibrating nuclei, and thus to assign appropriate effective masses. In any case, these effective mass corrections to the Born-Oppenheimer approximation are small in absolute magnitude but should be carefully considered when high-accuracy theory and/or calculations are contemplated.

Each of the Born-Oppenheimer potential functions Φ(r1 ... rN|n') can be viewed as defining a hypersurface embedded in a (3N + 1)-dimensional space that is generated by the 3N Cartesian coordinates of nuclear position supplemented by an energy axis. Because many of the basic interests of condensed matter science require N to be roughly comparable to Avogadro's number (≈ 6.022 × 1023), these hypersurfaces are enormously complicated. Nevertheless, mathematical description at least of a statistical sort is feasible for these hypersurfaces, and it is facilitated to some extent by analogies to the topographies of familiar three-dimensional landscapes. A substantial portion of the analyses presented in subsequent chapters exploits this viewpoint.

Although it proves to be an excellent description for many problems in condensed-matter physics and chemical physics, the Born-Oppenheimer approximation is inaccurate and thus inappropriate in some special circumstances. In addition to the effective-mass corrections mentioned above, one of these cases involves nuclear dynamics where close approach (in energy), or even intersection, of two or more potential energy hypersurfaces occurs [Yarkony, 1996, 2001; Domcke and Yarkony, 2012]. Another important case concerns superconductivity arising from the BCS mechanism of electron coupling through phonons [Bardeen, Cooper, and Schrieffer, 1957], and in fact the relevant deviations from the Born-Oppenheimer description underlie the nuclear isotopic-mass dependence of superconducting transition temperatures [Kittel, 1963, Chapter 8; Ashcroft and Mermin, 1976, Chapter 34].

B. Properties of Electronic Ground and Excited States

Although the number of electronic eigenstates indexed by n' in the Born-Oppenheimer approximation is infinite for all material systems of interest and includes states of unbounded excitation energy above the ground state, the principal focus of attention in the present volume is on the ground state itself, and the low-lying excited states. If the atoms comprised in a many-body system are geometrically well isolated from one another, the lowest lying electronic states are nearly degenerate (i.e., confined to very narrow bands) and have energies substantially determined by those of the separate atoms. Bound excited states for isolated atoms are limited above by that atom's ionization energy. Table I.1 presents the ionization energies for individual atoms of several elements, i.e., the energy difference between the electronic ground state and the lowest lying ionized continuum state for those atoms. The substantial variation among entries for the elements is an indication of their distinctive chemical properties.

As a configuration of many identical atoms, initially widely separated from one another, is uniformly compressed, the narrow bands of nearly degenerate energy levels of the many-atom system tend to broaden in a manner strongly dependent on the specific atomic elements involved. When brought to normal ambient-pressure crystal densities and configurations, different elements can become electronic insulators (large remaining band gap between ground and lowest lying excited electronic state), semiconductors (small but nonzero band gap), or metals (no gap) [Harrison, 1980]. But it is well to keep in mind that even atomic species that are normally regarded as insulators (e.g., the noble gases) eventually become metallic materials under sufficiently strong compression. A case in point is the element xenon, which has been reported to transform to a metallic ground state at a pressure of 132(5) GPa [Goettal et al., 1989].


Excerpted from Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena by Frank H. Stillinger. Copyright © 2016 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Meet the Author

Frank H. Stillinger is senior scientist in the Department of Chemistry at Princeton University. He is a member of the National Academy of Sciences.

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