The Born-Oppenheimer electronic wavefunction Φ BO R (r) picks up a topological phase factor ±1, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in R-space. We show that this topological quantity reverts to a geometric quantity e iγ if the geometric phase γ = Im ΦR|∇µΦR · dRµ is evaluated with the conditional electronic wavefunction ΦR(r) from the exact electron-nuclear factorization ΦR(r)χ(R) instead of the adiabatic function Φ BO R (r). A model of a pseudorotating molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides the first examples of induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current which cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wavefunction.

The Kondo zero bias anomaly of Co adatoms probed by scanning tunneling microscopy is known to depend on the height of the tip above the surface, and this dependence is different on different low index Cu surfaces. On the (100) surface, the Kondo temperature first decreases then increases as the tip approaches the adatom, while on the (111) surface it is virtually unaffected. These trends are captured by combined density functional theory and numerical renormalization group (DFT+NRG) calculations. The adatoms are found to be described by an S = 1 Anderson model on both surfaces, and ab initio calculations help identify the symmetry of the active d orbitals. We correctly reproduce the Fano lineshape of the zero bias anomaly for Co/Cu(100) in the tunneling regime but not in the contact regime, where it is probably dependent on the details of the tip and contact geometry. The lineshape for Co/Cu (111) is presumably affected by the presence of surface states, which are not included in our method. We also discuss the role of symmetry, which is preserved in our model scattering geometry but most likely broken in experimental conditions.

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The ground state energy of a system of electrons and nuclei is proven to be a variational functional of the conditional electronic density nR(r), the nuclear wavefunction χ(R) and an induced vector potential Aµ(R) and quantum geometric tensor Tµν (R) derived from the conditional electronic wavefunction ΦR(r) over nuclear configuration space, where r = r1, r2, . . . are electronic coordinates and R = R1, R2, . . . are nuclear coordinates. The ground state (nR, χ, Aµ, Tµν ) can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing an effective potential vs(r) that depends parametrically on R, (ii) the Schrödinger equation for χ(R) and (iii) Euler-Lagrange equations that determine Tµν . The theory is applied to the E ⊗ e Jahn-Teller model. The foundations of density functional theory (DFT) [1,2] are inextricably tied to the Born-Oppenheimer approximation. In DFT applications, e.g. electronic band structure calculations, it often suffices to treat the nuclei classically or to fix them to their equilibrium positions. Quantum nuclear effects such as tunneling, delocalization and zero-point energy are, however, relevant for several interesting problems, e.g. the phases of ice [3][4][5][6] and the local structure of water [7][8][9][10], and were recently reported to enable thermally-activated tunneling of protons through a graphene layer [11,12]. Some quantum nuclear effects can be included in DFT-based calculations by quantizing nuclear vibrations on the adiabatic ground state potential energy surface, but because such an approach relies on the Born-Oppenheimer approximation, it is not formally exact. When the nuclear variables and electron-nuclear coupling are treated exactly and fully quantum mechanically, the electrons feel, instead of the external potential v(r) of DFT, a "weighted" potential − i |χ(R)| 2 Z i e 2 /|r−R i |dR, modified by the delocalization of the nuclear probability density |χ(R)| 2 , but also additional interactions induced by nonadiabatic electron-nuclear correlations [13,14] not included in standard DFT functionals.Particularly in time-dependent processes such as photoinduced chemical bond dynamics [15], proton transfer in hydrogen-bonded systems [16], dissociative adsorption of H 2 on Pd(100) [17], and molecular processes involving conical intersections of Born-Oppenheimer (BO) potential energy surfaces [18], nonadiabatic and quantum nuclear effects may be significant. Mixed quantum-classical approaches, which couple quantum mechanical electrons to classical nuclear motion, usually adopt an effective single-particle description of the electrons, and DFT is often the only method capable of treating large systems of interest with sufficient accuracy.For the further development of theories capable of describing quantum nuclear effects in large systems, it would be useful to know whether it is in principle possible to include full quantum nuclear motion and electronicvibrational coupling while retaining a density-functional formulation of the electronic part of t...