We have implemented a constrained CASSCF(2,2) calculation so as to study thermal electron transfer between a chlorine ion and a cluster of lithium atoms of variable size (from 1 to 17). Our calculations illustrate how the geometry of the ground state-charge transfer state crossing point (as well as the strength of a diabatic coupling) can depend sensitively on the number of metal ions (i.e., the size of the cluster) and the relative positioning of the donor and acceptor. Thus, this set of calculations is an initial step toward understanding the transition from homogeneous to heterogeneous electron transfer. In the future, these constrained calculations should allow us to model still far larger systems, ideally opening up a pathway to study meaningful electrochemical phenomena.

We derive and implement the necessary equations for solving a dynamically weighted, state-averaged constrained CASSCF(2,2) wave function describing a molecule on a metal surface, where we constrain the overlap between two active orbitals and the impurity atomic orbitals to be a finite number. We show that a partial constraint is far more robust than a full constraint. We further calculate the system-bath electronic couplings that arise because, near a metal, there is a continuum (rather than discrete) number of electronic states. This approach should be very useful for simulating heterogeneous electron transfer and electrochemical dynamics going forward.

In order to study electron-transfer mediated chemical processes on a metal surface, one requires not one but two potential energy surfaces (one ground state and one excited state) as in Marcus theory. In this letter, we report that a novel, dynamically weighted, state-averaged constrained CASSCF(2,2) (DW-SA-cCASSCF(2,2)) can produce such surfaces for the Anderson impurity model. Both ground and excited state potentials are smooth, they incorporate states with a charge transfer character, and the accuracy of the ground state surface can be verified for some model problems by renormalization group theory. Future development of gradients and nonadiabatic derivative couplings should allow for the study of nonadiabatic dynamics for molecules near metal surfaces.

We implement a rare-event sampling scheme for quantifying the rate of thermally activated nonadiabatic transitions in the condensed phase. Our Quantum mechanics/molecular mechanics (QM/MM) methodology uses the recently developed Interface for NonAdiabatic QM/MM in Solvent (INAQS) package to interface an elementary electronic structure package and a popular open-source molecular dynamics software (GROMACS) to simulate an electron transfer event between two stationary ions in a solution of acetonitrile solvent molecules. Nonadiabatic effects are implemented through a surface hopping scheme, and our simulations allow further quantitative insight into the participation ratio of a solvent and the effect of ion separation distance as far as facilitating electron transfer. We also demonstrate that the standard gas-phase approaches for treating frustrated hops and velocity reversal must be refined when working in the condensed phase with many degrees of freedom. The code and methodology developed here can be easily expanded upon and modified to incorporate other systems and should provide a great deal of new insight into a wide variety of condensed phase nonadiabatic phenomena.

We test a set of multiconfigurational wavefunction approaches for calculating the ground state electron population for a two-site Anderson model representing a molecule on a metal surface. In particular, we compare (i) a Hartree Fock like wavefunction where frontier orbitals are allowed to be nonorthogonal versus (ii) a fully non-orthogonal configuration interaction wavefunction based on constrained Hartree-Fock states. We test both the strong and weak metal-molecule hybridization (Γ) limits as well as the strong and weak electron-electron repulsion (U) limits. We obtain accurate results as compared with exact numerical renormalization group theory, recovering charge transfer states where appropriate. The current framework should open a path to run molecular non-adiabatic dynamics on metal surfaces.

We present an efficient set of methods for propagating excited-state dynamics involving a large number of configuration interaction singles (CIS) or Tamm-Dancoff approximation (TDA) single-reference excited states. Specifically, (i) following Head-Gordon et al., we implement an exact evaluation of the overlap of singly-excited CIS/TDA electronic states at different nuclear geometries using a biorthogonal basis and (ii) we employ a unified protocol for choosing the correct phase for each adiabat at each geometry. For many-electron systems, the combination of these techniques significantly reduces the computational cost of integrating the electronic Schrodinger equation and imposes minimal overhead on top of the underlying electronic structure calculation. As a demonstration, we calculate the electronic excited-state dynamics for a hydrogen molecule scattering off a silver metal cluster, focusing on high-lying excited states, where many electrons can be excited collectively and crossings are plentiful. Interestingly, we find that the high-lying, plasmon-like collective excitation spectrum changes with nuclear dynamics, highlighting the need to simulate non-adiabatic nuclear dynamics and plasmonic excitations simultaneously. In the future, the combination of methods presented here should help theorists build a mechanistic understanding of plasmon-assisted charge transfer and excitation energy relaxation processes near a nanoparticle or metal surface.

We show that the Berry force as computed by an approximate, mean-field electronic structure can be meaningful if properly interpreted. In particular, for a model Hamiltonian representing a molecular system with an even number of electrons interacting via a two-body (Hubbard) interaction and a spin-orbit coupling, we show that a meaningful nonzero Berry force emerges whenever there is spin unrestriction-even though the Hamiltonian is real-valued and formally the on-diagonal single-surface Berry force must be zero. Moreover, if properly applied, this mean-field Berry force yields roughly the correct asymptotic motion for scattering through an avoided crossing. That being said, within the context of a ground-state calculation, several nuances do arise as far interpreting the Berry force correctly, and as a practical matter, the Berry force diverges near the Coulson-Fischer point (which can lead to numerical instabilities). We do not address magnetic fields here.

We analyze a model problem representing a multi-electronic molecule sitting on a metal surface. Working with a reduced configuration interaction Hamiltonian, we show that one can extract very accurate ground state wavefunctions as compared with the numerical renormalization group theory (NRG) -- even in the limit of weak metal-molecule coupling strength but strong intramolecular electron-electron repulsion. Moreover, we extract what appear to be meaningful excitation energies as well. Our findings should lay the groundwork for future {\em ab initio} studies of charge transfer processes and bond making/breaking processes on metal surfaces.

Bimetallic catalysts play important roles in the selective growth of single-walled carbon nanotubes (SWNTs). Using the simple salts (NH4)6W7O24·6H2O and Co(CH3COO)2·4H2O as precursors, tungsten-cobalt catalysts were prepared. The catalysts were composed of W6Co7 intermetallic compounds and tungsten-dispersed cobalt. With the increase of the W/Co ratio in the precursors, the content of W6Co7 was increased. Because the W6Co7 intermetallic compound can enable the chirality specified growth of SWNTs, the selectivity of the resulting SWNTs is improved at a higher W/Co ratio. At a W/Co ratio of 6 : 4 and under optimized chemical vapor deposition conditions, we realized the direct growth of semiconducting SWNTs with the purity of ∼96%, in which ∼62% are (14, 4) tubes. Using salts as precursors to prepare tungsten-cobalt bimetallic catalysts is flexible and convenient. This offers an efficient pathway for the large-scale preparation of chirality enriched semiconducting SWNTs.