Rapid calculation of Born-Oppenheimer (B-O) forces is essential for driving the so-called quantum region of a multi-scale molecular dynamics simulation. The success of density functional theory (DFT) with modern exchange-correlation approximations makes DFT an appealing choice for this role. But conventional Kohn-Sham DFT, even with various linear-scaling implementations, really is not fast enough to meet the challenge of complicated chemo-mechanical phenomena (e.g. stress-induced cracking in the presence of a solvent). Moreover, those schemes involve approximations that are difficult to check practically or to validate formally. A popular alternative, Car-Parrinello dynamics, does not guarantee motion on the B-O surface. Another approach, orbital-free DFT, is appealing but has proven difficult to implement because of the challenge of constructing reliable orbital-free (OF) approximations to the kinetic energy (KE) functional. To be maximally useful for multi-scale simulations, an OF-KE functional must be local (i.e. one-point). This requirement eliminates the two-point functionals designed to have proper linear-response behavior in the weakly inhomogeneous limit. In the face of these difficulties, we demonstrate that there is a way forward. By requiring only that the approximate functional deliver high-quality forces, by exploiting the "conjointness" hypothesis of Lee, Lee, and Parr, by enforcing a basic positivity constraint, and by parameterizing to a carefully selected, small set of molecules we are able to generate a KE functional that does a good job of describing various H q Si m O n clusters as well as CO (providing encouraging evidence of transferability). In addition to that positive result, we discuss several major negative results. First is definitive proof that the conjointness hypothesis is not correct, but nevertheless is useful. The second is the failure of a considerable variety of published KE functionals of the generalized gradient approximation type. Those functionals yield no minimum on the energy surface and give completely incorrect forces. In all cases, the problem can be traced to incorrect behavior of the functionals near the nuclei. Third, the seemingly obvious strategy of direct numerical fitting of OF-KE functional parameters to reproduce the energy surface of selected molecules is unsuccessful. The functionals that result are completely untransferable.

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The formulas of Fromm and Hill and of Remiddi, for the three-electron correlated atomic integral using Slater-type atomic orbitals, are shown to be equivalent in their common region of applicability. The demonstration required the derivation of an identity connecting dilogarithm functions which is either new or not widely known. It is then shown how to modify the more general of these formulas ͑that of Fromm and Hill͒ to eliminate the necessity of branch tracking in the complex plane and thereby to achieve a straightforwardly computable formalism. By obtaining a formal cancellation of the individual-term singularities in the Fromm-Hill formula, problems of numerical instability are also avoided. The work is validated with a computer program and sample calculations.

We present an analysis and extension of our constraint-based approach to orbital-free ͑OF͒ kinetic-energy ͑KE͒ density functionals intended for the calculation of quantum-mechanical forces in multiscale moleculardynamics simulations. Suitability for realistic system simulations requires that the OF-KE functional yield accurate forces on the nuclei yet be computationally simple. We therefore require that the functionals be based on density-functional theory constraints, be local, be dependent at most upon a small number of parameters fitted to a training set of limited size, and be applicable beyond the scope of the training set. Our previous "modified-conjoint" generalized-gradient-type functionals were constrained to producing a positive-definite Pauli potential. Though distinctly better than several published generalized-gradient-approximation-type functionals in that they gave semiquantitative agreement with Born-Oppenheimer forces from full Kohn-Sham results, those modified-conjoint functionals suffer from unphysical singularities at the nuclei. Here we show how to remove such singularities by introducing higher-order density derivatives and analyze the consequences. We give a simple illustration of such a functional and a few tests of it.

Erratum: Properties of constraint-based single-point approximate kinetic energy functionals [Phys. Rev. B 80, 245120 (2009)]

We describe an expansion method for calculating scattering phase shifts which avoids the difficulties of the variational methods of Hulthen 1 and Kohn. 2 Demkov and Shepelenko 3 have pointed out the algebraic source of these difficulties, and they have shown how the Kohn and Hulthen methods implicitly eliminate an equation to restore consistency. These methods have also been examined elsewhere. 4 ? 5 A third variational method, due to Schwinger, 6 appears to be more difficult to use, although Schwartz 7 has indicated a way to apply Schwinger's method for potential scattering, and Lippmann 8 has recently converted the Schwinger principle into a form similar to those of Hulthen and Kohn. Other approaches are those of Schlessinger and Schwartz, 9 and of Spruch, Rosenberg, Hahn, and O'Malley. 10 The former proceeds by analytic continuation from boundstate regions, while the latter simplifies calculations for compound targets.Ours is an approach which seeks to use to advantage the singularity properties that adversely affect the Kohn method. As it stands, it is applicable to compound targets, and it appears extendable to multichannel processes. Consider a single-channel process with Hamiltonian H and asymptotic solutions ip t (E) and $ 2 (E) at energy E. The stationary-state wave function^ is then written \£ = a 1 $ l + a 2 il) 2 + , thereby defining $, and the Schrodinger equation yields aAH-E)il>AE)+a (H-E)ipJE) + (H-E) = 0. (I) 1 1 z IWe assume H to be such that at large distances the incident particle sees a short-range potential, so that (H-E)r\> ly (F-E)^2, (H-E)$, and $ are all quadratically integrable. The solution of Eq. (1) for $ then should be expected to be smoothly approximable with boundstate functions. Introduce a set of bound-state functions X(, i-1, • • * 9 n, to be used for the approximation of $. The role of these functions is most clearly seen if they are transformed to a basis on which H is diagonal within the subspace spanned by the x;-That is, we construct and solve the finite matrix equation (H-A §) c = 0, where Hy = (xflXj) and S t j = (xtXj)-The solution 5^, cor-responding to eigenvalue A ", defines a function (P^TJV 0 viiXp* Now, suppose that the functions cp ^ are capable of giving a good representation of $ at an energy E. Then Eq. (1) should be nearly satisfied. The condition we shall impose here is that the left side of Eq. (1) have no component in the subspace spanned by the cp . Because of the bounded nature of all terms of Eq.(1), this condition should produce convergence to a well-defined solution as the cp " approach a complete set of quadratically integrabe functions.The essence of our method is to recognize that the above-described condition is easy to impose if E is chosen to be an eigenvalue A" of the finite probelm which we used to define the cp ". For E = X the coefficient of cp " on the left of Eq. (1) is fl l^'%'* 1 V >+8 2 < % l % , W > '

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The various theories of polyelectrolyte behavior are examined and the nature of the approximations inherent in each are discussed. A new model of the polyion in solution is proposed and an outline is described for the evaluation of the free energy of the polyelectrolyte solution, including the effects of dissociation, binding and entropy of distribution of the unionized groups, ions and bound ion pairs. The binding of counter ions by the polymer is related to the pH, and comparison with experimental data of the theoretical relationship gives indications as to the nature of the binding process. The entropy of distribution among charged and uncharged sites on the polymer chain is evaluated, considering interaction between nearest neighboring sites, and is found to decrease markedly from its value for a random distribution, especially at intermediate degrees of neutralization. This large decrease in entropy as the degree of neutralization increases is responsible for the extensive binding of counter ions which is experimentally observed. These manifestations of non-ideal behavior make it impossible to use the previously accepted relationship between the pH of the solution and the thermodynamic properties of the polyion.