Scaling for Electronic Structure

It was argued above that many chemical systems exhibit some degree of localization. What this means is that if one atom moves by a small amount, the electrons and nuclei relax in a way that screens out, over some length scale, the effects of that movement. This concept can be quantified by looking at the one-electron density matrix. In a Kohn-Sham calculation, the electron density at a point r is given by 

The total energy in Hartree-Fock theory or DFT can be expressed entirely in terms of this density matrix. It can be shown that, for systems with a band gap (that is, a separation between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) energies) or for 

Two apparent drawbacks of the localized-orbital approach exist. First, the convergence of the total energy stalls at a value above the exact numerical solution depending on the size of the cutoff radius for the orbitals. This makes sense because physical information is lost in the truncation. Second, the convergence rate appears to slow somewhat with increasing system size. The observed convergence rates are still good, and competitive with other numerical methods, but this slowdown does not fit with standard MG orthodoxy—the 

---

Efforts to tame the unfavorable scaling of electronic structure methods are not limited to density functional theory. For a general summary of the current state of the art see the review by Goedecker, 1999. 

Mauri, F., G. Galli, and R. Car. 1993. Orbital formulation for electronic-structure calculations with linear system-size scaling. Phys. Rev. B 47, 9973. 

In order to obtain estimates of quantum transport at the molecular scale [105], electronic structure calculations must be plugged into a formalism which would eventually lead to observables such as the linear conductance (equilibrium transport) or the current-voltage characteristics (nonequilibrium transport). The directly measurable transport quantities in mesoscopic (and a fortiori molecular) systems, such as the linear conductance, are characterized by a predominance of quantum effects—e.g., phase coherence and confinement in the measured sample. This was first realized by Landauer [81] for a so-called two-terminal configuration, where the sample is sandwiched between two metalhc electrodes energetically biased to have a measurable current. Landauer s great intuition was to relate the conductance to an elastic scattering problem and thus to quantum transmission probabilities. 

A quantitative scale for the structural effect of various silyl groups is established, as shown in entry 57 of Table 1, by the rates of solvolysis of 40 triorganosilyl chlorides in aqueous dioxane under neutral conditions69. The structural effect involves the steric effect and, in some examples, the electronic effect. Because little difference exists in the electronic effect among alkyl groups, their steric effect at silicon follows the order primary < secondary < tertiary substituents. 

Zheng, J. Alecu, I. M. Lynch, B. J. Zhao, Y Tmhlar, D. G. Database of frequency scaling factors for electronic structure methods, 2003, URL http //comp.chem. umn.edu/freqscale/index.html... 

The inappropriate scaling of the RLDA with Z, and thus also with j6, becomes particularly obvious for fixed electron number. In Fig. 5.5 the percentage deviations of the RLDA for and Ej are shown for the Ne isoelectronic series. The error for the correlation energy in the RLDA shows little tendency to approach zero with increasing Z, indicating that the relativistic correction factor plotted in Fig. 4.4 is inadequate for electronic structure calculations. 

In spite of this interest of physical and biochemists, prior to the work of Hansch and Fujita there appears to have been no eifort to devise a scale of hydrophobicity analogous to the Hammett scale for electronic effects, or the Taft scale for steric effects. Part of this reluctance was due to the selection of the relevant solvent system for such a scale. In addition, biochemists who were interested in hydrophobic stabilization of protein structures needed only to consider the few hydrophobic amino acid side chains. 

On the other hand, there is an obvious demand for electronic-structure methods from first principles which are able to treat huge chemical systems composed of, say, several thousands of atoms. Let us think of nano- and biomaterials, just to name two important research directions. Unfortunately, standard methods will not do here simply because their scaling behavior with respect to the number of atoms is insufficient. We recall that the time needed to diagonalize a DFT-based Hamiltonian goes with the third power of the size of the system. In the real world, mathematical tricks may fortunately be used to bring this cubic scaling down to something of the order of Nonethe-... 

In the Z — 1 limit, the kinetic terms involving derivatives remain, the centrifugal terms drop out, and the scaled Coulombic potentials metamorphize into delta functions. This hyperquantum limit is tantamount to — oo in the unsealed wave equation. For electronic structure, the low-D limit is generally less useful than the large-D limit, because only the ground state of a delta-function potential [2,5,9] is bound and that can accommodate only two electrons. However, for... 

The classical, Marcus/Hush, limit corresponds to Equation (1) with /Cei= 1 and = (FC)r=o-This condition is achieved if either (i) the structural differences between the reactants and products do not implicate high-frequency vibrational modes or (ii) the exchange of energy (heat) between the high-frequency vibrational modes and the solvent is fast on the time scale for electron transfer. Statement (ii) is equivalent to the equilibrium assumption of transition-state theory. That this assumption is not always correct for reactions in solution has been demonstrated in ultrafast kinetic studies of reactions that vary ... 

Greeley BH et al (1994) New pseudospectral algorithms for electronic-structure calculations -length scale separation and analytical 2-electron integral corrections. J Chem Phys 101(5) 4028-4041... 

Harrison, et al. have reported an efficient, accurate multiresolution solver for the Kohn-Sham and Hartree-Fock self-consistent field methods for general polyatomic molecules. The Hartree-Fock exchange is a nonlocal operator, whose evaluation has been a computational bottleneck for electronic structure calculations, scaling as for small molecules... 

At least three major scales must be distinguished in simulations of heterogeneous media (i) the atomistic scale, required to account for electronic structure effects in catalytic systems or for molecular and hydrogen bond fluctuations that govern the transport of protons and water (ii) the scale of the electrochemical double layer, ranging from several A to a few nm at this level, simulations should account for potential and ion distributions in the metal-electrolyte interfacial region and (iii) the scale of about 10 nm to 1 pm, to describe transport and reaction in heterogeneous media as a function of composition and porous structure. 

Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

AMI Combined Quantum Mechanical and Molecular Mechanical Potentials Combined Quantum Mechanics and Molecular Mechanics Approaches to Chemical and Biochemical Reactivity Hybrid Methods Hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) Methods Linear Scaling Methods for Electronic Structure Calculations Mixed Quantum-Classical Methods MNDO MNDO/d Parameterization of Semiempirical MO Methods PM3 Quantum Mechanical/Molecular Mechanical (QM/MM) Coupled Potentials Quantum Mechanics/Molecular Mechanics (QM/MM) Semiempirical Methods Integrals and Scaling. 

Linear Scaling Methods for Electronic Structure Calculations... 

LINEAR SCALING METHODS FOR ELECTRONIC STRUCTURE CALCULATIONS 1497... 

---