Chapter 2. Theoretical background
2.5 Molecular properties
Once the solution of the Schrödinger equation has been approximated by any of the methods in Chapter 2.2, one can turn to molecular property calculations. One way of expressing molecular properties is in the language of analytical derivatives. We wish to see how the energy changes when a small perturbation is applied and so we
∆Aa→b= � ξb ξa � dE dξ � ξ dξ = − � ξb ξa � f(ξ)�ξdξ
k
=
k
BT
h
e
−∆G‡ RTperform a Taylor expansion of the energy with respect to the perturbation. This involves calculating derivatives of the energy, sometimes resulting in complicated expressions for each of the terms of the energy functional used to solve the
Schrödinger equation, especially if second derivatives or higher are required.8 The
following subchapters discuss the electric field gradient (a first order property, where only the first derivative of the energy is required) and the chemical shift (a second order property, where the second derivative of the energy must be calculated).
2.5.1 Electric field gradients (EFG)
The interaction between a quadrupolar nucleus (a nucleus with spin I > ½) and its environment, can be written as an operator:60
(42) where e is the elementary charge, Q(A) is the electric quadrupole moment of nucleus A,
I(A) is the spin-‐orbit operator of nucleus A and the F(A) is the field gradient operator of
nucleus A. The matrix element of the electronic field-‐gradient operator is defined as:60
(43) where μ and ν are the Cartesian components (μ,ν {x,y,z}), δμν is the delta function and riA is the distance between electron i and nucleus A.
The electronic contribution to the EFG tensor V is directly calculated using the available density matrix from a converged SCF calculation :8
(44) where V(A)μν, ele is the second-‐rank EFG tensor, Pκτ is the density matrix (the matrix of all optimised basis set coefficients from the SCF calculation) and φ are the basis functions.
The nuclear contribution is then calculated by:61
(45) a classical equation as the nuclei are approximated as static point charges. The total EFG tensor is finally V = V(A)μν, ele + V(A)μν, nuc .
The full EFG tensor, V, can be represented as a matrix in the Cartesian coordinate system:62 (46) HQ= e2 � A,i Q(A)I(A)F(A)(i)I(A) Fµν(A)(i) = r 2 iAδµν−3riA;µriA;ν r5 iA Vµν,ele(A) =� κ,τ Pκτ�ϕκ|r−A5(r2Aδµν −3rA;µrA;ν)|ϕτ� Vµν,nuc(A) = � Y�=A ZY r2 Aδµν−3rA;µrA;ν r5A V = � � � � � � Vxx Vxy Vxz Vyx Vyy Vyz Vzx Vzy Vzz � � � � � �
The EFG tensor is symmetric (Vxy = Vyx and so on) and can additionally be
diagonalised (i.e expressing it in the principal axis system), which makes the off-‐ diagonal elements vanish and results in the three principal components (or eigenvalues), here called Vxx, Vyy, Vzz and three eigenvectors, describing the
orientation of the tensor in space. The EFG is non-‐zero only when the charge distribution around the nucleus violates cubic symmetry. The largest of the
eigenvalues (here defined to be Vzz ) is then used to calculate the nuclear quadrupole
coupling constant (NQCC) :
(47) and the asymmetry parameter is defined as follows:
(48) where Vzz ≥ Vxx ≥Vyy .
These two parameters are the only ones (other equations involving the product of Q and Vzz are also sometimes used) needed to fully describe the quadrupole interaction
as it occurs in spectra due to the trace of the tensor being zero (Vzz + Vyy +Vxx = 0 ). The
NQCC and the asymmetry parameter are usually refined during line shape analysis of solid-‐state NMR spectra as well as gas-‐phase microwave (MW) spectra.
The EFG tensor is usually calculated in atomic units (au) by quantum chemistry codes. We note that different sign conventions are in use, e.g. the one used in the Gaussian 0363/0964 programs requires multiplication by -‐1 in order to get the correct
sign for the NQCC for comparison with experimental data (when the sign of the NQCC is available).62
In our work, Gaussian-‐style Vzz values are always reported but CQ values from
calculations have always been converted to the correct sign (the experimental sign). The following equation is used for convenience:
(49) where Vzz is in au, CQ is in MHz, Q is in fm2 and k = -‐2.34964781 au-‐1 Bohr2 fm-‐2 MHz, is
a constant that takes care of unit conversion, includes both e and h and takes care of the correct sign use for CQ.
2.5.2 NMR chemical shifts
NMR-‐active nuclei are those with nuclear spin I > 0 and have 2I+1 nuclear energy levels, all degenerate under no applied field. When a field is applied, the degeneracy is lifted, opening up the possibility of performing spectroscopy by exciting the nucleus from one energy level to the other.
The resonance frequency is the frequency of electromagnetic radiation that will excite the nucleus to another energy level and is defined as:
(50) CQ= eQVzz h ηQ = Vxx−Vyy Vzz CQ=kQVZZ ν= γB0 2π (1−σ)
where ν is the resonance frequency, γ the gyromagnetic ratio of the NMR nucleus
(ratio of the magnetic dipole moment and its angular momentum) for each nuclide, B0
the applied magnetic field and σ the shielding. The shielding results in the effective magnetic field at the nucleus to be different than the applied field. In NMR
experiments, the resonance frequency is thus dependent on both the applied field and the shielding. Since the interest lies in using the magnetic resonance as a probe to chemical properties, the shift in resonance frequency is reported relative to a reference compound resulting in field-‐independent chemical shifts, δ.
The experimental chemical shift, δ, is therefore directly related to the shielding:
(51) The last approximation seen in the Eq. 51 can be made because σref is usually very
small compared to unity.
From a non-‐relativistic quantum mechanical perspective, the shielding, or the magnetic shielding tensor σNij of a nucleus N, is the mixed second derivative of the total molecular energy E with respect to the magnetic moment μN and the external
magnetic field B:65
(52) Computationally, the shielding is usually calculated as a sum of diamagnetic and paramagnetic contributions:8 (53) It is the paramagnetic part σ(p)ij that is slightly troublesome to compute as it involves the derivative of the density matrix. This necessitates the use of the coupled-‐
perturbed SCF equations making chemical shift calculations more expensive than most other properties.8
A further complication arises because the magnetic operators involved depend on the origin of the coordinate system leading to artificial origin-‐dependent results. This is called the gauge problem. If a complete basis set is used, the results will be
independent of the chosen origin, which is of course never the case in practice. Different ways of dealing with this gauge-‐dependency have been proposed, the most satisfactory solution probably being the use of magnetic-‐field dependent basis sets, the so called gauge-‐including atomic orbitals method (GIAO).66 With this method, the
basis functions are made magnetic-‐field dependent by multiplying them by a phase factor referring to the position of the nucleus:8
(54) δ= νsample−νref νref × 106 = σref−σsample 1−σref ≈ σref −σsample σNij =�δBiδµδ2EN j � B,µN=0 σij =σ(ijd)+σ (p) ij σij(d)= α 2 2 � µν Pµν�ϕµ|r−A3[rAr−rA,irj]|ϕν� σ(p)ij = α 2 2 � µν δPµν δBi � ϕµ|(−i∇ ×(r−RA))jr−A3|ϕν� ˜ ϕµ(χ,B) =ϕAµexp(i(α/2)(B×RA)r)
The shielding is, due to its mixed second derivative form, an unsymmetrical tensor with 3 × 3 different elements.
(55) The matrix can be split up into a symmetric part and an antisymmetric part. The antisymmetric part typically does not influence NMR spectra and is thus not of
interest. The symmetric part of the tensor, σsym = ½ (σ + σt ), where σt is the transpose
of σ, can be diagonalised if expressed in the principal axis system, giving three orthogonal eigenvectors and three eigenvalues, σ11, σ22 and σ33 that describe the
position and shape of the NMR signal (ignoring J coupling, quadrupole coupling and other effects). The eigenvectors describe the orientation of the tensor components in space.62
The average of the eigenvalues is the isotropic shielding, σiso = ⅓ (σ11 + σ22 + σ33).
The isotropic chemical shift, δiso, of a nucleus in a molecule is evaluated by taking the
difference in the isotropic shielding between it and the nucleus in an accepted
reference molecule (usually the one that defines the origin of the chemical shift scale).
(56) Computational chemical shifts are hence evaluated in an analogous way to NMR experiments.
The principal components of the chemical shift tensor, that can be obtained from a solid-‐state NMR spectrum, can be represented in several different ways. In this work we follow the Haeberlen-‐Mehring-‐Spiess convention67-‐69 as shown in Eqs. 57-‐60 and
Figure 1. (57) (58) (59) (60) where δσ is called the reduced anisotropy and ησ the asymmetry parameter.
σ= � � � � � � σxx σxy σxz σyx σyy σyz σzx σzy σzz � � � � � �
δ
iso=
σ
ref−σ
isoδ
ii=
σ
iso,(ref)−σ
ii(i
= 1,2,3)
|δ
zz−δ
iso| ≥ |δ
xx−δ
iso| ≥ |δ
yy−δ
iso|
δ
σ=δ
zz−δ
isoησ = δyy−δxx
Figure 1 A static solid-‐state NMR spectrum showing a specific assignment of the principal components according to the Haeberlen-‐Mehring-‐Spiess convention. Under magic angle spinning (MAS) conditions this spectrum would turn into a spinning-‐sideband pattern with an envelope similar to this figure.
Using this convention, the isotropic chemical shift gives the position of the NMR signal on the chemical shift scale (a relative shielding) while the reduced anisotropy and the asymmetry parameter describe the shape of the signal relative to the
isotropic chemical shift.
Finally, the spin-‐spin coupling constant is a second order property like the chemical shift and is defined as the second derivative of the energy w.r.t. the magnetic
moments of the nuclei. It involves the separate computation of four different terms that together contribute to the coupling pattern that often shows up in NMR
spectra.70 These are the Fermi contact term (KFC), the spin dipole term (KSD), the
paramagnetic spin-‐orbit term (KPSO) and the diamagnetic spin orbit term (KDSO), as
originally deduced by Ramsey.71
Computation of NMR properties usually involves single-‐point calculations on previously optimised geometries using any quantum mechanics based method for which the necessary molecular property integrals have been coded. Normal basis sets can often be used, although it has been found that the addition of tight (large
exponent) p functions for shielding constants72 and tight s, p, d and f functions for
spin-‐spin couplings73 can sometimes improve results.