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Register a new userLocal representation of N-body Coulomb energy with path integrals
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We represent N-body Coulomb energy in a localized form to achieve massive parallelism. It is a well-known fact that Greens functions can be written as path integrals of field theory. Since two-body Coulomb potential is a Greens function of Poisson equations, it reduces to a path integral of free scalar field theory with three spatial dimensions. This means that N-body one also reduces to a path integral. We discretize real space with a cubic lattice and evaluate the obtained multiple integrals approximately with the Markov-chain Monte Carlo method.
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An expression for the Green function G(E;x_1,x_2) of the Schroedinger equation is obtained through the approximations of the path integral by n-fold multiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have peaks near the values of the energy levels E_{j}. The analytic and numerical examples for one-dimensional and multi-dimensional harmonic and anharmonic oscillators, and Poeschl-Teller potential wells, show that median values of these peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to the exact values of even levels already in the lowest orders of approximation n=1 and n=2, i.e. when the path integral is replaced by a line or double integral. The weights of the peaks approximate the values of the squared modulus of the wave functions at x=0 with the same accuracy.
We study the decomposition of the Coulomb integrals of periodic systems into a tensor contraction of six matrices of which only two are distinct. We find that the Coulomb integrals can be well approximated in this form already with small matrices compared to the number of real space grid points. The cost of computing the matrices scales as O(N^4) using a regularized form of the alternating least squares algorithm. The studied factorization of the Coulomb integrals can be exploited to reduce the scaling of the computational cost of expensive tensor contractions appearing in the amplitude equations of coupled cluster methods with respect to system size. We apply the developed methodologies to calculate the adsorption energy of a single water molecule on a hexagonal boron nitride monolayer in a plane wave basis set and periodic boundary conditions.
Whenever variables $phi=(phi^1,phi^2,ldots)$ are discarded from a system, and the discarded information capacity $mathcal{S}(x)$ depends on the value of an observable $x$, a quantum correction $Delta V_mathrm{eff}(x)$ appears in the effective potential [arXiv:1707.05789]. Here I examine the origins and implications of $Delta V_mathrm{eff}$ within the path integral, which I construct using Synges world function. I show that the $phi$ variables can be `integrated out of the path integral, reducing the propagator to a sum of integrals over observable paths $x(t)$ alone. The phase of each path is equal to the semiclassical action (divided by $hbar$) including the same correction $Delta V_mathrm{eff}$ as previously derived. This generalises the prior results beyond the limits of the Schrodinger equation; in particular, it allows us to consider discarded variables with a history-dependent information capacity $mathcal{S}=mathcal{S}(x,int^t f(x(t))mathrm{d} t)$. History dependence does not alter the formula for $Delta V_mathrm{eff}$.
Deep learning has fostered many novel applications in materials informatics. However, the inverse design of inorganic crystals, $textit{i.e.}$ generating new crystal structure with targeted properties, remains a grand challenge. An important ingredient for such generative models is an invertible representation that accesses the full periodic table. This is challenging due to limited data availability and the complexity of 3D periodic crystal structures. In this paper, we present a generalized invertible representation that encodes the crystallographic information into the descriptors in both real space and reciprocal space. Combining with a generative variational autoencoder (VAE), a wide range of crystallographic structures and chemistries with desired properties can be inverse-designed. We show that our VAE model predicts novel crystal structures that do not exist in the training and test database (Materials Project) with targeted formation energies and band gaps. We validate those predicted crystals by first-principles calculations. Finally, to design solids with practical applications, we address the sparse label problem by building a semi-supervised VAE and demonstrate its successful prediction of unique thermoelectric materials
The Breit correction, the finite-light-speed correction for the Coulomb interaction of the electron-electron interaction in $ O left( 1/ c^2 right) $, is introduced to density functional theory (DFT) based on the non-relativistic reduction with the local density approximation. Using this newly developed relativistic DFT, it is found that the possible outer-most electron of lawrencium atom is the $ p $ orbital instead of the $ d $ orbital, which is consistent with the previous calculations based on wave-function theory. A possible explanation of the anomalous behavior of its first ionization energy is also given. This DFT scheme provides a practical calculation method for the study of properties of super-heavy elements.
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