SUPPORTING INFORMATION
Proton Reduction at Surface of Transition Metal Nanocatalysts
Qingguo Meng, Jiangchao Chen, Dmitri Kilin*
Department of Chemistry, University of South Dakota, 414E Clark, Vermillion, SD 57069
COMPUTATIONAL DETAILS
Density functional theory with the generalized gradient approximation (GGA)1 implemented in
the Vienna ab initio simulation package (VASP)2 is employed for the structural optimization and
total energy calculations. Total energy 𝐸𝑇𝑂𝑇 [𝜌𝛼 , 𝜌𝛽 ] is expressed in terms of 𝜌𝛼 (𝑟⃑) and 𝜌𝛽 (𝑟⃑)
which are the spin-up and spin-down electron densities, corresponding to the diagonal elements in
spin space, and which play the same role as total electron density, 𝜌(𝑟⃑), does in the absence of a
magnetic field.3 The electron density 𝜌(𝑟⃑), and spin density 𝜁(𝑟⃑) of the Kohn-Sham (KS) orbitals
for a given spin component can be expressed as in equation (S1) and (S2) respectively.
𝜌(𝑟⃑) = 𝜌𝛼 + 𝜌𝛽
𝜁(𝑟⃑) =
(S1)
𝜌𝛼 − 𝜌𝛽
(S2)
𝜌
Then, the Kohn-Sham (KS) equation in spin polarization can be expressed as
ħ2
𝐾𝑆
𝐾𝑆
(− 2𝑚 ∇2 + 𝑣𝛼 [ 𝑟⃑, 𝜌𝛼 (𝑟⃑)])𝜑𝑖𝛼
(𝑟⃑) = 𝜀𝑖𝛼 𝜑𝑖𝛼
(𝑟⃑)
(−
ħ2 2
∇ +
2𝑚
𝐾𝑆
𝐾𝑆
𝑣𝛽 [ 𝑟⃑, 𝜌𝛽 (𝑟⃑)])𝜑𝑖𝛽
(𝑟⃑) = 𝜀𝑖𝛽 𝜑𝑖𝛽
(𝑟⃑)
(S3)
(S4)
where the first and second terms in each equation correspond to kinetic energy ( 𝑇𝛼,𝛽 ) and
𝐾𝑆
𝐾𝑆
potential energy ( 𝑣𝛼,𝛽 ), 𝜑𝑖𝛼
( 𝑟⃑) and 𝜑𝑖𝛽
( 𝑟⃑) are the KS spin orbitals, and 𝜀𝑖𝛼 and 𝜀𝑖𝛽 are
eigenenergies. The total energy functional 𝐸𝑇𝑂𝑇 [𝜌𝛼 , 𝜌𝛽 ] does exist for given change densities
𝜌𝛼 (𝑟⃑) and 𝜌𝛽 (𝑟⃑) , and includes three interactions: coulomb, exchange, and correlation. The
potential can be expressed as a functional derivative of the exact energy functional
𝐸𝑇𝑂𝑇 [𝜌𝛼 , 𝜌𝛽 ] − 𝑇 with respect to variation of the spin component density (S5) and (S6), where T
represents kinetic energy in equation (S3) and (S4).
𝑣𝛼 [ 𝑟⃑, 𝜌𝛼 (𝑟⃑)] =
𝑣𝛽 [ 𝑟⃑, 𝜌𝛽 (𝑟⃑)] =
𝛿(𝐸𝑇𝑂𝑇 [𝜌𝛼 , 𝜌𝛽 ]−𝑇𝛼 )
𝛿𝜌𝛼 (𝑟⃑)
𝛿(𝐸𝑇𝑂𝑇 [𝜌𝛼 , 𝜌𝛽 ]−𝑇𝛽 )
𝛿𝜌𝛽 (𝑟⃑)
(S5)
(S6)
The total density of electrons is the combination of partial charge densities of KS orbitals with
𝐾𝑆
𝐾𝑆
known spin |𝜑𝑖𝛼
( 𝑟⃑)|2 or |𝜑𝑖𝛽
(𝑟⃑)|2 . Each of these contributions are taken with occupation
number 𝑓𝑖𝛼 and 𝑓𝑖𝛽 , which take on values of 0 and 1.
𝐾𝑆
𝜌𝛼 (𝑟⃑) = ∑𝑖 𝑓𝑖𝛼 |𝜑𝑖𝛼
(𝑟⃑)|2
(S7)
𝐾𝑆
𝜌𝛽 (𝑟⃑) = ∑𝑖 𝑓𝑖𝛽 |𝜑𝑖𝛽
(𝑟⃑)|2
(S8)
Equations (S3)-(S8) are solved in an iterative manner in VASP. The orbital character is analyzed
by projecting the wavefunctions onto spherical harmonics within spheres of a Wigner–Seitz
radius around each ion4. The output contains the decomposition of hybrid molecular orbitals into
the form of s, p d, and f orbital contributions from each ion to each hybrid molecular orbital.
It is important to note that the total number of electrons can be different for spin α and β,
as represented in the following equations, which is the basic concept for constrained DFT 5.
𝑁𝛼 = ∫ 𝜌𝛼 𝑑𝑟 = ∑𝑖 𝑓𝑖𝛼
(S9)
𝑁𝛽 = ∫ 𝜌𝛽 𝑑𝑟 = ∑𝑖 𝑓𝑖𝛽
(S10)
The sum of (𝑁𝛼 + 𝑁𝛽 ) is fixed and equals the total number of electrons.
BACKGROUND CHARGE IN DFT:
The convergence of the electrostatic energy in calculations using periodic boundary
conditions is considered in the context of periodic solids and localized aperiodic systems in
the gas and condensed phases. Conditions for the absolute convergence of the total energy
in periodic boundary conditions are obtained, and their implications for calculations of the
properties of polarized solids under the zero-field assumption are discussed. For aperiodic
systems the exact electrostatic energy functional in periodic boundary conditions is
obtained. The convergence in such systems is considered in the limit of large supercells,
where, in the gas phase, the computational effort is proportional to the volume. It is shown
that for neutral localized aperiodic systems in either the gas or condensed phases, the
energy can always be made to converge as O(L-5) where L is the linear dimension of the
supercell. For charged systems, convergence at this rate can be achieved after adding
correction terms to the energy to account for spurious interactions induced by the periodic
boundary conditions. These terms are derived exactly for the gas phase and heuristically for
the condensed phase.
Figure S1. DOS of Pd13H24 ̶ (left) and Pd13H24+ (right) clusters.
1.
Juan, Y. M.; Kaxiras, E.; Gordon, R. G., Use of the Generalized Gradient
Approximation in Pseudopotential Calculations of Solids. Physical Review B 1995, 51 (15),
9521-9525.
2.
Hafner, J., Ab-initio simulations of materials using VASP: Density-functional theory
and beyond. Journal of Computational Chemistry 2008, 29 (13), 2044-2078.
3.
Yang, R. G. P. a. W., Density -Functional Theory of Atoms and Molecules. Oxford
University Press: 1994.
4.
(a) Girifalco, L. A., Statistical Mechanics of Solids Oxford University Press: 2000; (b)
Politzer, P.; Parr, R. G.; Murphy, D. R., Approximate determination of Wigner-Seitz radii from
free-atom wave functions. Physical Review B 1985, 31 (10), 6809-6810.
5.
Kaduk, B.; Kowalczyk, T.; Van Voorhis, T., Constrained Density Functional Theory.
Chemical Reviews 2011, 112 (1), 321-370.