2.7.1. Effects of Relativity in Atoms
In atomic units, the radial velocity of the innermost electrons in an atomic system is approximately equal to the charge on the nucleus. For atoms with Z > 40, this radial velocity is a significant proportion of the speed of light, c, and relativistic effects become non-negligible.
Relativistic motion of the innermost electrons is partially responsible for the way that heavy elements behave, with a notable example being the yellow colouring of gold234,235: Non-relativistic calculations of the excitation energies from the 5d to 6s levels for gold predict transition bands at high energies, in the UV region, but relativistic motion of the inner electrons affects the energies of electron orbitals. When relativistic effects are included, transitions are seen at lower energies, in the middle of the visible region, accounting for the distinctive yellow colour of the metal. This demonstrates the importance of appropriately treating relativistic effects when dealing with heavy elements232.
Two different components of relativistic effects can be defined: scalar relativistic effects or effects which are due to the spin-orbit coupling. Scalar effects are caused by the high velocities of inner electrons in heavy atoms. This leads to a relativistic mass increase,
𝑚𝑒 = 𝑚𝑜
√1 − 𝑣𝑐22
(𝐸𝑞. 2.29)
Where 𝑚𝑜 is the rest mass of the electron, 𝑣 is its radial velocity and 𝑐 is the speed of light. Where 𝑣 is a significant percentage of 𝑐 (true, in general, for elements heavier than Zirconium), this leads to a contraction of the inner orbitals, and an increased stabilisation of the s and p orbitals. The contraction of these orbitals increases the screening effect of the attractive force the nucleus enacts on the d and f electrons, causing them to become destabilised and extended.
67 Spin-orbit coupling effects arise due to the interaction between the magnetic field generated by an electron orbiting the atomic nucleus and the electron’s spin. Russell-Saunders coupling is applicable when the spin-orbit coupling is weak, and has the effect of splitting orbitals into pairs according to the rule
𝑗 = 𝑙 ± 1
2 (𝐸𝑞. 2.30)
Where j is the angular momentum quantum number and 𝑙 is the orbital angular momentum quantum number. In a given system, for each value of 𝑙 there are two orbitals, each with a differing j value.
2.7.2. Relativistic Hamiltonians
Relativistic effects can be accounted for in quantum chemical calculations by modifying the Hamiltonian to include scalar relativistic and/or spin-orbit coupling terms. This section will briefly discuss methods for inclusion of these terms, beginning with the Dirac equation (𝐸𝑞. 2.31) which attempts to account for relativistic effects for a single electron304. The time-independent Dirac equation is:
[𝑐𝜶 ∙ 𝑃̂ + 𝑐2𝜷 + 𝑉]𝛹 = 𝐸𝛹 (𝐸𝑞. 2.31)
With the Dirac Hamiltonian, where α and β are both matrices:
ℎ̂D = 𝑐𝜶 · 𝑃̂ + 𝑐2𝜷 + 𝑉 (𝐸𝑞. 2.32)
𝜶𝑥,𝑦,𝑧 = ( 0 𝝈𝑥,𝑦,𝑧
𝝈𝑥,𝑦,𝑧 0 ) , 𝜷 = (𝐈 0
0 𝐈) (𝐸𝑞. 2.33)
𝑃̂ is the momentum operator, 𝑐 is the speed of light and 𝑉 is a potential. 𝝈𝑥,𝑦,𝑧 are the three 2 × 2 Pauli spin matrices and I is a 2 × 2 unit matrix. This equation describes an electron-positron pair, with both spin states of each accounted for explicitly, hence the four component wavefunction:
Ψ(𝒙) = [Ψ1(𝒙), Ψ2(𝒙), Ψ3(𝒙), Ψ4(𝒙)] (𝐸𝑞. 2.34)
68 The Dirac equation is very computationally expensive compared to solving the non-relativistic Schrödinger equation and as it only describes a single electron-positron pair, it is not useful for molecular calculations.
For a molecular calculation, a generalisation of the Dirac equation to a many-particle system is needed. The Dirac-Coulomb-Breit (DCB) Hamiltonian is one such generalisation of the Dirac Hamiltonian to an N-particle system305,306. It accounts for both scalar and spin-orbit relativistic effects and despite being initially proposed in 1928, remains the most accurate way of including relativistic effects in quantum chemical calculations:
𝐻̂𝐷𝐶𝐵 = ∑ ℎ̂𝑖
𝑖
+ ∑ ℎ̂𝑖𝑗
𝑖<𝑗
(𝐸𝑞. 2.35)
Where ℎ̂𝑖is the Dirac Hamiltonian and ℎ̂𝑖𝑗 is the two-particle term:
ℎ̂𝑖𝑗 = 1
𝒓𝑖𝑗 + 1
2𝒓𝑖𝑗 [𝜶𝑖 · 𝜶𝑗+ (𝜶𝑖 · 𝒓𝑖𝑗)(𝜶𝑗 · 𝒓𝑖𝑗)
𝒓𝑖𝑗2 ] (𝐸𝑞. 2.36)
Applying the Dirac-Coulomb-Breit Hamiltonian to a four-component wavefunction as in the Dirac equation is significantly more computationally expensive than solving the non-relativistic Schrodinger equation for the same system.
Decoupling the scalar relativistic effects from the relativistic effects due to the spin orbit coupling allows the latter to be neglected, if desired. The zeroth-order regular approximation (ZORA) is one such method307–311. ZORA is a zeroth-order perturbational expansion of the Dirac equation312:
Ĥ𝑠𝑐𝑎𝑙𝑎𝑟𝑍𝑂𝑅𝐴 + Ĥ𝑠𝑝𝑖𝑛−𝑜𝑟𝑏𝑖𝑡𝑍𝑂𝑅𝐴 = 𝑉 + 𝛔 · 𝑃̂ 𝑐2
2𝑐2− 𝑉𝛔 · 𝑃̂ (𝐸𝑞. 2.37)
Where 𝑃̂ is the momentum operator, V is a potential and σ is the spin-orbit matrix.
69 It is desirable to eliminate the positronic (negative energy) states present in the Dirac Hamiltonian. The Douglas-Kroll-Hess (DKH) Hamiltonian arises from a unitary transformation of the Dirac Hamiltonian313–315, splitting it into two parts, with one part describing electrons and the other describing the positronic negative-energy states.
Decoupling the negative and positive energy terms results in an infinite series of operators (𝐸𝑞. 2.38), where 𝜀̂𝑘 are the expansion terms. The lower orders of this series can be used to account for relativistic effects in quantum calculations in a computationally efficient and accurate manner316.
ℎ̂𝐷𝐾𝐻∞ = ∑ 𝜀̂𝑘
∞
𝑘=0
(𝐸𝑞. 2.38)
Most quantum chemical calculations can be performed sufficiently with a second order expansion, so the expansion can be truncated at k=2, and the Hamiltonian is then known as DKH2314,317.
2.7.3. Relativistic Pseudopotentials
As discussed in Chapter 2.3: Basis Sets, pseudopotentials present a way to reduce computational expense while treating relativistic effects implicitly285,236. Only the valence electrons are likely to be involved in chemical processes, so a heavier atom can be split into a core region, comprising the nucleus and the inner electrons which can then be treated implicitly with a relativistic pseudopotential (see Chapter 2.3: Basis Sets), and the valence electrons, which are treated explicitly.
To generate a relativistic pseudopotential, an all-electron wavefunction for the atom must be generated using a relativistic Hamiltonian38,285,287,318. As discussed previously, the valence orbitals are then replaced with one-electron pseudo-orbitals, which are fitted to the valence all-electron orbitals by parameterisation of the potential.
Pseudopotentials designed for the treatment of f-block elements can treat the f electrons explicitly or model them as part of the core, which simplifies the calculations but reduces accuracy284,287,318–322.
70