Quantum molecular spin

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

Photons in quantum optical cavities also constitute excellent qubit candidates [52]. Resonant coupling of atoms with a single mode of the radiation field was experimentally achieved 25 years ago [53], and eventually the coherent coupling of quantum optical cavities with atoms or (simple) molecules was suggested as a means to achieve stable quantum memories in a hybrid quantum processor [54]. There might be a role to play for molecular spin qubits in this kind of hybrid quantum devices that combine solid-state with flying qubits. 

A wide variety of proof-of-principle systems have been proposed, synthesized and studied in the field of molecular spin qubits. In fact, due to the fast development of the field, several chemical quantum computation reviews using magnetic molecules as spin qubits have been published over the past decade, covering both experimental and theoretical results [67-69]. Only in a minority of experiments implementing non-trivial one- or two-qubit gates has been carried out, so in this aspect this family is clearly not yet competitive with other hardware candidates.1 Of course, the main interest of the molecular approach that makes it qualitatively different is that molecules can be chemically engineered to tailor their properties and acquire new functionalities. 

For a pulse-type NMR experiment, the assumption has a straightforward interpretation, since the pulse applied at the moment zero breaks down the dynamic history of the spin system involved. The reasoning presented here, which leads to the equation of motion in the form of equation (72), bears some resemblance to Kaplan and Fraenkel s approach to the quantum-mechanical description of continuous-wave NMR. (39) The crucial point in our treatment is the introduction of the probabilities izUa which are expressed in terms of pseudo-first-order rate constants. This makes possible a definition of the mean density matrix pf of a molecule at the moment of its creation, even for complicated multi-reaction systems. The definition of the pf matrix makes unnecessary the distinction between intra- and inter-molecular spin exchange which has so far been employed in the literature. 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

Figure 13. Short-time relaxation in dimmers (a) field dependence of the short-time square-root relaxation rates are presented on a logarithmic scale showing the depletion of the molecular spin states by quantum tunneling at H g = 0.42 T for various waiting times tdig and (b) difference between the relaxation rate in the absence and in the presence of digging, fhole = fsqrt(0) - rsqrt(tdig).

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable... 

These effects on NMR can either be described by a classical treatment of the demagnetization field or by a macroscopic quantum treatment which removes several important simplifications such as the concept of the molecular spin system and the rejection of the high-temperature approximation. However, Levitt has noted that only the classical treatment produces quantitative agreement with experiment. 

We begin our discussion of wave function based quantum chemistry by introducing the concepts of -electron and one-electron expansions. First, in Sec. 2.1, we consider the expansion of the approximate wave function in Slater determinants of spin orbitals. Next, we introduce in Sec. 2.2 the one-electron Gaussian functions (basis functions) in terms of which the molecular spin orbitals are usually constructed the standard basis sets of Gaussian functions are finally briefly reviewed in Sec. 2.3. 

The spin is not affected by the axial electric field, thus the atomic spins, Si and S2, couple to form the molecular spin quantum number as... 

In this chapter, we reviewed different quantum chemical approaches to determine local quantities from (multireference) wave functions in order to provide a qualitative interpretation of the chemical bond in open-shell molecules. Chemical bonding in open-shell systems can be described by covalent interactions and electron-spin coupling schemes. For different definitions of the (effective) bond order as well as various decomposition schemes of the total molecular spin expectation value into local contributions, advantages and shortcomings have been pointed out. For open-sheU systems, the spin density distribution is an essential ingredient in the... 

Building wave functions in molecular quantum chemistry starts from generating a basis set, which normally consists of atom-centered functions obtained from ealeulations on individual atoms. Let us assume that there are m of these atomic orbitals (AOs) and call them (Xp,P= )-We can construct the MOs, pt as a linear expansion in these basis functions (the LCAO method LCAO = linear combination of atomic orbitals). A spin function has to be attached to each molecular orbital and we define the molecular spin orbitals, (SO) as ... 

We have placed special emphasis on using a consistent notation throughout the book. Since quantum chemists use a number of different notations, it is appropriate to define the notation we have adopted. Spatial molecular orbitals (with latin indices ij, k...) are denoted by These are usually expanded in a set of spatial (atomic) basis functions (with greek indices ju, V, A,...) denoted by 0. Molecular spin orbitals are denoted by x Occupied molecular orbitals are specifically labeled by a, b, c,... and unoccupied (virtual) molecular orbitals are specifically labeled by r, s, r,... Many-electron operators are denoted by capital script letters (for example, the Hamiltonian is Jf), and one-electron operators are denoted by lower case latin letters (for example, the Fock operator for electron-one is /( )). The exact many-electron wave function is denoted by O, and we use T to denote approximate many-electron wave functions (i.e., the Hartree-Fock ground state wave function is o while FS is a doubly excited wave function). Exact and approximate energies are denoted by S and , respectively. All numerical quantities (energies, dipole moments, etc.) are given in atomic units. 

Since the orientations of the molecular spin momenta are quantized, it is natural to treat the spin waves by a quantum mechanical theory. This theory follows closely the methods outlined in the previous subsection. First, the ground state of the spin system... 

---