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Scaling Up Accurate Electronic Structure Calculations by Patching Exchange-Correlation Potential in Materials Chen Huang Department of Scientific Computing, Florida State University Email: [email protected] Background and Motivation Computer simulations have become invaluable in materials re- search for solving emerging problems in clean energy, nan- otechnology, catalysis, etc. To understand and predict the prop- erties of materials, accurate electronic structures are the pre- requisites. The many-body Schr¨ odinger equation is difficult to solve for large-scale materials. Density functional theory (DFT) is the workhorse for material research nowadays, but is severely limited by the approximated exchange-correlation (XC) functionals. In this work, I derive a general way to ob- tain reliable electronic structure in materials by constructing accurate XC potential.[1] With an accurate XC potential, the Kohn-Sham (KS) DFT becomes accurate. Step 1. Partition Materials We start with selecting one atom (call it atom-q ) in a mate- rial. Some nearby atoms are then selected as its buffer atoms. The atom and its buffer atoms form the cluster, with the rest atoms being the environment. We give an initial guess of the KS potential V tot KS of the material, which yields the total elec- tron density by solving the Kohn-Sham equation. Cluster is defined by projecting V tot KS to its region as V cluster KS ( ~ r )= w cluster ( ~ r )V tot xc ( ~ r ) Environment’s KS potential is defined as V env KS = V tot KS - V cluster KS . The weight w cluster is designed to satisfy a key re- quirement for the partitioning that V cluster KS becomes equal to V tot KS inside the cluster. Figure 1: (a) The model N6 molecule. The nitrogen atoms are equally spaced. (b,c) The parititioned electron densities of the N2-N3-N4 cluster and its environment (N1, N5, and N6). The N2 and N4 are the buffer atoms of the N3. We use an embedding potential V emb to “glue” the cluster with its environment. By adjusting the embedding potential, the sum of n cluster ( ~ r ) and n env ( ~ r ) becomes equal to the to- tal density. We demonstrate the density decomposition with a model N 6 molecule (Fig. 1). The cluster consists of atoms “N2”, “N3”, and “N4”, and the rest nitrogen atoms form the environment. Cluster and environment’s densities are very lo- calized, which is a must for keeping cluster’s Hilbert space as small as possible. Step 2: Computing Cluster’s XC potential For cluster density n cluster ( ~ r ), we solve for its XC potential V cluster xc by inverting the Schr¨ odinger equation. We solve for the external potential V ( ~ r ) that yields n cluster ( ~ r ). This can be done by constructing the W e functional W e = E [V ] - Z V ( ~ r )n cluster ( ~ r )dr 3 , where E [V ] is the energy of the cluster with V as an additional external potential. Once we have maximized W e , V ( ~ r ) yields the cluster density. The gradient for maximizing W e can be easily derived. Cluster’s XC potential is obtained as V cluster xc ( ~ r )= V emb ( ~ r )+ V cluster KS ( ~ r ) - V ( ~ r ) - V H [n cluster ]( ~ r ) We then project cluster’s XC potential to the atom-q . Step 3: Patching XC Potential in Entire Material Steps 1 and 2 are performed for all the atoms in the materi- als. By stitching the high-level XC potentials of all atoms, we obtain the high-level XC potential of the entire material. We then go back to Step 1 to complete the self-consistent loop. A flowchart is demonstrated (Fig. 2). Our method can be consid- ered as a local optimized effective potential (OEP) problem, which consists of two steps: (a) partitioning and (b) patching. The partitioning step can be efficiently parallelized in practice. Figure 2: The flowchart of solving the N 6 model molecule using the XC- patching method. We have six “cluster-environment” pairs, since we have six N atoms. Outlook and Acknowledgment We are building a software based on the XC-patching method developed here, with the aim to predict electronic structure of large-scale (strongly) correlated materials completely based on computer simulations in a parameter-free way. We are grateful for the financial support from the Florida State University. References [1] Chen Huang. Journal of Chemical Theory and Computa- tion. doi:10.1021/acs.jctc.6b00051 (2016) in press.
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Scaling Up Accurate Electronic Structure Calculations byPatching Exchange-Correlation Potential in Materials
Chen HuangDepartment of Scientific Computing, Florida State University
Email: [email protected]
Background and Motivation
Computer simulations have become invaluable in materials re-search for solving emerging problems in clean energy, nan-otechnology, catalysis, etc. To understand and predict the prop-erties of materials, accurate electronic structures are the pre-requisites. The many-body Schrodinger equation is difficultto solve for large-scale materials. Density functional theory(DFT) is the workhorse for material research nowadays, butis severely limited by the approximated exchange-correlation(XC) functionals. In this work, I derive a general way to ob-tain reliable electronic structure in materials by constructingaccurate XC potential.[1] With an accurate XC potential, theKohn-Sham (KS) DFT becomes accurate.
Step 1. Partition Materials
We start with selecting one atom (call it atom-q) in a mate-rial. Some nearby atoms are then selected as its buffer atoms.The atom and its buffer atoms form the cluster, with the restatoms being the environment. We give an initial guess of theKS potential V tot
KS of the material, which yields the total elec-tron density by solving the Kohn-Sham equation. Cluster isdefined by projecting V tot
KS to its region as
V clusterKS (~r) = wcluster(~r)V
totxc (~r)
Environment’s KS potential is defined as V envKS = V tot
KS −V clusterKS . The weight wcluster is designed to satisfy a key re-
quirement for the partitioning that V clusterKS becomes equal to
V totKS inside the cluster.
Figure 1: (a) The model N6 molecule. The nitrogen atoms are equallyspaced. (b,c) The parititioned electron densities of the N2-N3-N4 clusterand its environment (N1, N5, and N6). The N2 and N4 are the buffer atomsof the N3.
We use an embedding potential Vemb to “glue” the clusterwith its environment. By adjusting the embedding potential,the sum of ncluster(~r) and nenv(~r) becomes equal to the to-tal density. We demonstrate the density decomposition witha model N6 molecule (Fig. 1). The cluster consists of atoms“N2”, “N3”, and “N4”, and the rest nitrogen atoms form theenvironment. Cluster and environment’s densities are very lo-calized, which is a must for keeping cluster’s Hilbert space assmall as possible.
Step 2: Computing Cluster’s XC potential
For cluster density ncluster(~r), we solve for its XC potentialV clusterxc by inverting the Schrodinger equation. We solve for
the external potential V (~r) that yields ncluster(~r). This can bedone by constructing the We functional
We = E[V ]−∫
V (~r)ncluster(~r)dr3,
where E[V ] is the energy of the cluster with V as an additionalexternal potential. Once we have maximized We, V (~r) yieldsthe cluster density. The gradient for maximizing We can beeasily derived. Cluster’s XC potential is obtained as
V clusterxc (~r) = Vemb(~r) + V cluster
KS (~r)− V (~r)− VH[ncluster](~r)
We then project cluster’s XC potential to the atom-q.
Step 3: Patching XC Potential in Entire Material
Steps 1 and 2 are performed for all the atoms in the materi-als. By stitching the high-level XC potentials of all atoms, weobtain the high-level XC potential of the entire material. Wethen go back to Step 1 to complete the self-consistent loop. Aflowchart is demonstrated (Fig. 2). Our method can be consid-ered as a local optimized effective potential (OEP) problem,which consists of two steps: (a) partitioning and (b) patching.The partitioning step can be efficiently parallelized in practice.
Figure 2: The flowchart of solving the N6 model molecule using the XC-patching method. We have six “cluster-environment” pairs, since we havesix N atoms.
Outlook and Acknowledgment
We are building a software based on the XC-patching methoddeveloped here, with the aim to predict electronic structure oflarge-scale (strongly) correlated materials completely based oncomputer simulations in a parameter-free way. We are gratefulfor the financial support from the Florida State University.
References
[1] Chen Huang. Journal of Chemical Theory and Computa-tion. doi:10.1021/acs.jctc.6b00051 (2016) in press.
