Auxiliary Density Matrix Method (ADMM)

The previously described method helps reduce computational cost by using the GPW ap-proach. However, hybrid functional and extensive basis sets both greatly increase compute time. Furthermore, the cost of computing the HFX term scales to the fourth power with respect to the number of primitive basis set functions used. Fortunately, as system size increases many of these integrals can be screened to reduce computational cost. If heavily contracted basis sets are used, the number of primitive integrals increases. In addition, the number of functions required grows quadratically with quantum number making po-larization functions incredibly expensive. Finally, diffuse functions further compound the issue since they reduce the region of space where electron density is negligible. This makes screening more difficult, further increasing the expense of calculating HFX. In order to use hybrid functionals while minimizing BSSE, the efficiency of calculating HFX must be improved.

To accomplish this, I employed the Auxiliary Density Matrix Method (ADMM) [17].

When using an atomic centered basis set φ_µ(r) we can write

ψ_i(r) =X

µ

C^µiφ_µ(r). (2.9)

The Hartree-Fock exchange component can then be written as a product of the density matrix and the two electron integrals as shown below.

E_X^{HF X}[P ] = −1 2

X

λσµν

P^µσP^νλ(µν|λσ). (2.10)

The density matrix elements P^µν can be obtained using the molecular (MO) coefficients C as follows:

P^µν =X

i

C^µiC^νi, (2.11)

and the two electron integrals can be represented as:

(µν|λσ) = ( Z Z

φ_µ(r₁)φ_ν(r₁)g(|r₂− r₁|)φ_λ(r₂)φ_σ(r₂)dr₁dr₂). (2.12) In this equation, g(r) represents the (1/r) Coulombic interaction potential in standard Hartree-Fock theory. The density matrix P can then be replaced by an auxiliary density matrix ˆP that is either smaller or rapidly decaying and the HFX energy term can be written as follows:

E_x^{HF X}[P ] = E_x^{HF X}[ ˆP ]+(E_x^{HF X}[P ]−E_x^{HF X}[ ˆP ]) ≈ E_x^{HF X}[P ]+(E_x^{DF T}[P ]−E_x^{DF T}[ ˆP ]). (2.13)

In this scheme, the critical approximation is that the difference between exchange energy calculated using the primary and auxiliary density matrices is well represented by GGA calculations. As the auxiliary basis set becomes identical to the primary basis set, this difference converges to zero. However, the rate of convergence depends not only on the primary basis set, but the functional as well. The above equation states that a correction factor based on the difference in the exchange term between using primary or auxiliary density matrices must be applied when computing the HFX energy using an auxiliary density matrix. Since only a small fraction of exchange is used in most hybrid functionals, any error in the final calculation will be greatly reduced. The current implementation of the correction term is based on the GGA correction for PBE exchange [5, 6]. Using other parameterizations and functionals for the correction term has not been explored.

The ADMM greatly reduces the computational cost of combining hybrid functionals with extensive basis sets. This allowed me to minimize BSSE while ensuring that the elec-tronic structure of my systems were properly represented with reasonable computational cost.

Auxiliary Basis Sets

In order to use the ADMM, an auxiliary density matrix must be generated. This can be done using a variety of methods. These range from directly manipulating the sparsity of the matrix during the calculation to generating it from less extensive auxiliary basis sets.

In this work, I have elected to generate auxiliary density matrices from auxiliary basis sets fit to model systems. Since the primary density matrix P generated using a high quality primary basis set (φ_mu(r)) is large, an auxiliary basis set ( ˆφ_mu(r)) is used to describe the wave function during HFX calculations instead:

ψ_i(r) =X

µ

Cˆ^µiφˆ_µ(r). (2.14)

The auxiliary density matrix elements ˆP^µν are obtained from molecular coefficients as follows:

Pˆ^µν =X

i

Cˆ^µiCˆ^νi. (2.15)

The optimal value for these MO coefficients are then obtained by minimizing the square of the difference between occupied wave functions in the primary and auxiliary basis set representations: Auxiliary basis sets for the first two rows of elements are provided with the original documentation [17]. However, for the work presented in Chapters 5 and 6, I needed to generate additional auxiliary basis sets in order to account for Ni and Co atoms.

Basis Set Parameterization

The first step to generating auxiliary basis sets for the missing atom types was to choose the functional form. For simplicity, a set of 10 Gaussian type orbitals was selected as

the auxiliary basis set along with several small Ni containing systems to use as a fit-ting set. The auxiliary basis set chosen contains four S functions, three P functions and three D functions. They are designed to be used with the GTH pseudopotentials [18]

and the MOLOPT primary basis set [19]. Ni atoms in this work were represented using 18 valence electrons (3s²4s²3p⁶4d⁸) and the Co atoms were represented using 17 valence electrons (3s²4s²3p⁶4d⁷). The fitting environments included a small NiO cluster (8 atoms), C4H6N iO4, N i4O4, N iCO3, and N i(OH)2 as shown in Figure 2.1.

Figure 2.1: A set of small systems used to generate the auxiliary basis set for Ni and another set of systems chosen to evaluate transferability and accuracy of the auxiliary basis set are shown. Ni atoms are shown in brown, O atoms in red, C atoms in blue, and H atoms in white.

The geometries of these systems were optimized using the primary (MOLOPT) basis set in CP2K to obtain minimized geometries. The coefficients of the auxiliary basis set were then optimized to minimize the energy at these predetermined geometries with the auxiliary basis set. The optimized coefficients for each fitting system are given below in Table 2.1, along with the coefficients optimized using a single Ni atom. The similarities between these optimized auxiliary basis sets indicate good transferability between systems of a similar chemical environment. However, the results obtained from a single Ni atom show that the 10 Gaussian form selected may not be sufficient to represent an isolated

atom.

Table 2.1: A table of the auxiliary basis set coefficients generated using the four fitting systems and a single Ni atom. The auxiliary basis sets generated from each of the fitting systems are nearly identical. When fitting onto the neutral Ni atom, however, there is a large difference in one of the P functions which is highlighted in bold. These values represent the exponents of the Gaussians and are given in Bohr⁻².

I then averaged the parameters obtained over the four fitting environments to produce transferrable auxiliary basis set for Ni. This auxiliary basis set was then evaluated using a set of four evaluation systems including two larger NiO clusters (27 and 36 atoms), Ni with four adjacent water molecules, and Ni(CO)₄, as shown in Figure 2.1. The energy difference between calculations performed with and without enabling the ADMM was found to be less than 10 meV. A more detailed examination reveals that even the one electron energy levels within these simple systems is well represented using the ADMM, with differences in the range of 0.1 to 0.05 eV (see Supporting Table A.1).

This process was then repeated for Co. Additionally, I included an auxiliary basis set that was directly fit to the Co-Salen molecule for comparison, as shown in Table 2.2. As with the Ni atom, the isolated Co atom shows some minor differences when compared to the Co-Salen and fitting systems.

I then used these auxiliary basis sets in conjunction with the ADMM to perform DFT calculations of CO molecules and Co-Salen molecules on NiO(100) in chapters 5 and 6.

Examples files for parametrizing auxiliary basis sets can be found on the supplementary disk attached to this thesis.

Co-Salen Fitting Set Co Atom

Table 2.2: A table containing optimized auxiliary basis set parameters for Co. They were generated from fitting directly to the Co-Salen molecule in addition to the set of fitting molecules for comparison. When fitting to the isolated Co atom, however, there is again a large difference in one of the P functions which is highlighted in bold. This is similar to the difference observed for the Ni atom. These values represent the exponents of the Gaussians and are given in Bohr⁻².