Guido Fano

Quantum chemistry using the density matrix renormalization group

A new implementation of the density matrix renormalization group is presented for ab initio quantum chemistry. Test computations have been performed of the dissociation energies of the diatomics Be2, N2, HF. A preliminary calculation on the Cr2 molecule provides a new variational upper bound to the ground state energy.

Quantum chemistry using the density matrix renormalization group II

We have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Moller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the...

The density matrix renormalization group method. Application to the PPP model of a cyclic polyene chain.

The density matrix renormalization group (DMRG) method introduced by White for the study of strongly interacting electron systems is reviewed; the method is variational and considers a system of localized electrons as the union of two adjacent fragments A,B. A density matrix ρ is introduced, whose eigenvectors corresponding to the largest eigenvalues are the most significant, the most probable states of A in the presence of B; these states are retained, while states corresponding to small eigenvalues of ρ are neglected. It is conjectured that the decreasing behavior of the eigenvalues is Gaussian. The DMRG method is tested on the Pariser-Parr-Pople Hamiltonian of a cyclic polyene (CH)N up to N=34. A Hilbert space of dimension 5.×1018 is explored. The ground state energy is 10−3 eV within the full CI value in the case N=18. The DMRG method compares favorably also with coupled cluster approximations. The unrestricted Hartree-Fock solution (which presents spin density waves) is briefly reviewed, and a compar...

Density matrix renormalization group study of dimerization of the Pariser–Parr–Pople model of polyacetilene

We apply the density matrix renormalization group method to the Pariser–Parr–Pople Hamiltonian and investigate the onset of dimerization. We deduce the parameters of the hopping term and the contribution of the σ bonds from ab initio calculations on ethylene. Denoting by Rij the C–C distances, we perform a variational optimization of the dimerization δ=(Ri,i+1−Ri−1,i)/2 and of the average bond length R0 for chains up to N=50 sites. The critical value of N at which the transition occurs is found to be between N=14 and N=18 for the present model. The asymptotic values for large N for R0 and δ are given by 1.408(3) A and 0.036(0) A.

Mathematical Methods in Quantum Mechanics

The mathematical methods used in quantum mechanics are developed, with emphasis on linear algebra and complex variables. Dirac notation for vectors in Hilbert space is introduced. The representation of coordinates and momenta in quantum mechanics is analyzed and applied to the Heisenberg uncertainty principle.

Twentieth-Century Quantum Mechanics

The historical development and fundamental principles of quantum mechanics are reviewed. Fundamental differences with classical statistical mechanics are emphasized. The representation of quantum phenomena in Hilbert space is introduced.

Digital and Quantum Computers

The fundamentals of digital and quantum computers are presented. Binary numbers, modular arithmetic, and Boolean algebra are reviewed. Logic gates for both classic and quantum computers are introduced and combined in some simple circuits for carrying out useful computations. The possibility for efficient factoring of large numbers using quantum computers is discussed in detail. This requires a lengthy detour into number theory. The most obvious application is to cryptography and secure communication.

The Schrödinger Equation

The Schrodinger equation is introduced and applied to some elementary problems: particle-in-a box, harmonic oscillator, angular momentum, and the hydrogen atom. The representation of eigenfunctions and eigenvalues is discussed, employing linear operators and matrices. The quantum theory of spin, as well as the Pauli exclusion principle, are described. These enable a theoretical understanding of atomic structure and the periodic table. The two disparate modes of time-dependence of a quantum system–unitary evolution and collapse of the wavefunction–are contrasted.

New Adventures: Isotropic Vectors, Rotations, Spinors, and Groups

Some more specialized mathematical topics are introduced, including isotropic vectors, rotations, spinors, and Lie groups. The concept of invariance in the objective world is discussed. The stereographic projection is introduced to describe the behavior of spinors. The Lie groups SO(3) and SU(2) are studied in detail.

Quantum Entanglement and Bell’s Theorem

A description of quantum entanglement and its applications. Bell’s inequalities and Bell’s theorem are described, along with their implications for local reality and hidden variables. Other topics: applications using electron spin and photon polarization, Aspect’s experiments, decoherence of quantum states.