Nonadiabatic potential-energy surfaces by constrained density-functional theory

Nonadiabatic effects play an important role in many chemical processes. In order to study the underlying nonadiabatic potential-energy surfaces (PESs), we present a locally constrained density-functional theory approach, which enables us to confine electrons to subspaces of the Hilbert space, e.g., to selected atoms or groups of atoms. This allows one to calculate nonadiabatic PESs for defined charge and spin states of the chosen subsystems. The capability of the method is demonstrated by calculating nonadiabatic PESs for the scattering of a sodium and a chlorine atom, for the interaction of a chlorine molecule with a small metal cluster, and for the dissociation of an oxygen molecule at the Al(111) surface.

Figure 1

Matrix representation of the projection operators ${\mathbf{P}}_A^{\mathbf{k}}$ and ${\mathbf{P}}_B^{\mathbf{k}}$ in the Hilbert space of the Bloch basis functions ${\chi }_j^{\mathbf{k}}$. The $S_{ij}^{\mathbf{k}}$ are the overlap matrix elements between basis functions $i$ and $j$. $n$ is the total number of basis functions and $m$ is the number of basis functions of subsystem $A$.

Figure 2

(Color online) Nonadiabatic PESs for the scattering of a Na and a Cl atom. Shown are the energies as a function of the interatomic distance $Z$. See Table and the text for the constrained electron numbers defining the various nonadiabatic PESs. Additionally shown are the adiabatic ground state PES and the PES obtained with a FSM triplet calculation. The energy zero corresponds to the energy of the two isolated, neutral atoms.

Figure 3

(Color online) Level energy of the highest occupied Kohn-Sham molecular orbital (HOMO) of a free Na atom and of the lowest unoccupied Kohn-Sham molecular orbital (LUMO) of a Cl atom as a function of the electronic occupation. The occupations in the two separate calculations of the isolated atoms are varied in the form of a charge transfer, i.e., the Na atom is computed with a fraction of an electron removed, and the Cl atom is computed with this fraction of an electron added.

Figure 4

Force acting on the Cl atom in the NaCl dimer as a function of the interatomic distance $Z$ for the ionic PES. Filled squares represent the forces calculated within the LC-DFT approach and open triangles represent the forces resulting from a numerical differentiation of the PES shown in Fig. 2.

Figure 5

(Color online) Nonadiabatic PESs of a ${\mathrm{Cl}}_2$ molecule scattering at a ${\mathrm{Mg}}_4$ cluster. Shown are the energies as a function of the distance $Z$ of the centers-of-mass of the ${\mathrm{Cl}}_2$ molecule and the ${\mathrm{Mg}}_4$ cluster, with a scattering geometry as explained in the inset. See Table and the text for the constrained electron numbers defining the various nonadiabatic PESs. Additionally shown is the adiabatic ground-state PES, and the energy zero corresponds to the energy of the isolated neutral ${\mathrm{Cl}}_2$ molecule and ${\mathrm{Mg}}_4$ cluster.

Figure 6

(Color online) Two-dimensional cut (“elbow plot”) through the high-dimensional PES for the ${\mathrm{O}}_2$ dissociation at Al(111). The energy is shown as a function of the center-of-mass distance of the molecule from the surface $Z$ and the oxygen-oxygen bond length $r$. The molecule approaches the surface head-on above an fcc site as explained in the insets. (a) Adiabatic calculation. (b) Triplet fixed-spin moment calculation. (c) Neutral triplet LC-DFT calculation. Only the latter PES exhibits an energy barrier. The energy difference between the contour lines is $0.2\mathrm{eV}$ and the small white circle denotes the molecular position discussed in Fig. 7.

Figure 7

(Color online) Magnetization density (difference between spin-up and spin-down electron density) for a molecule at the energy barrier of the LC-DFT triplet PES ($r=1.4\mathrm{\AA }$, $Z=1.8\mathrm{\AA }$, as marked in Fig. 6). Shown is a two-dimensional cut perpendicular to the surface and through the ${\mathrm{O}}_2$ molecule, along the $\left[01\overline{1}\right]$ and [111] direction. The position of the two O atoms are marked as small black circles. The position of the Al atoms are marked as small white circles. (a) Isolated ${\mathrm{O}}_2$ molecule in its spin-triplet ground state, (b) adiabatic calculation for ${\mathrm{O}}_2∕\mathrm{Al}\left(111\right)$, (c) triplet FSM calculation for ${\mathrm{O}}_2∕\mathrm{Al}\left(111\right)$, and (d) neutral triplet LC-DFT calculation for ${\mathrm{O}}_2∕\mathrm{Al}\left(111\right)$.