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Concepts in Magnetic Resonance

Density Matrices in NMR Spectroscopy: Part II

Thomas C. Farrar

Published Apr 1, 1990• 21 citations
Publisher

Abstract

Abstract Density matrix theory for a weakly coupled two‐spin system is outlined. It is shown that the sixteen basis matrices needed to describe events in the two‐spin system are all simple direct products of pairs of the single‐spin matrices.
DOI
10.1002/cmr.1820020202

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The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.
Abstract The spin-independent properties of an electronic system are completely determined by spinless density functions, P1(1; 1') and P 2( 1,2; 1',2 '), derived from the 1-and 2-electron density matrices, p1(1; 1') and p2(1,2 ; 1',2'). These functions, and their physical inter­-pretation, have been dealt with in part I: the present paper contains the generalizations necessary when the Hamiltonian contains spin terms (as in discussions of electron and nuclear spin resonance). The density matrices, and more generally the transition density matrices p1(kλ|1; 1') and p2(kλ|l,2; 1',2') associated with states and Φkand Φλ are resolved into spinless components associated with different spin situations: p1 has four such components (one giving the density of a-spin electrons, a second the density of β-spin electrons, and two with less direct physical significance), while p2 has sixteen components. It is shown that all components, for states k=(S, M),λ= (S',M') of given S, S', are determined by three 2-electron functions.For k=λ these reduce to p2, the spinless pair function, from which the electron density p1 may be derived by integration; Q2a ‘conditional' spin density, from which the (1-electron) spin density Q1 may be derived (as in the case of p1 and P2); Q c. a., a ‘coupling anisotropy function', which describes the anisotropy of spin-spin coupling associated with quantization of the spin z-component. It is also useful to introduce (though it is not an independent function). Qc, a ‘coupling density function', which measures the coupling between the spins of two electrons at different points in space. All spin-dependent effects, including the usual spin-spin and spin-orbit couplings and also (indirect) nuclear and electron-nuclear spin-spin couplings, may be related to the many- electron wave function through the above density functions. All the densities have a physical interpretation, independent of the precise nature of the wave functions (which may, for example, be the exact eigenfunctions of a spinless Hamiltonian or merely approximations of orbital form). The derivation of the density functions from approximate wave functions is considered in some detail. For functions describing weakly coupled electron groups (e. g. inner-shell and valence electrons) the densities may be expressed in terms of those of the separate groups. This reduction may be followed by an orbital description of each group; and subsequently, by a linear-combination-of-atomic-orbitals (l. c. a. o.) approximation. In the simpler approxi­mations, all the coupling effects discussed are determined by electron and spin ‘populations'. Preliminary calculations suggest that such approximations may in many cases be adequate.
A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw signal intensities of the single and double quantum transitions for the I=1 spin system are derived. Secondly a spin system with I=3/2 is described as a simple extension of the use of the Ii−jp operators. The excitation and the detection of the triple quantum coherence are discussed for noninteracting nuclei in solids. Finally the double quantum effects in a system of two interacting spins with I=1/2 are discussed.
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.