Introduction to Computational Quantum Chemistry
Lesson 11 : Relativistic Calculations
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 1
Why Relativistic Quantum Chemistry?
recall the non-relativistic Hamiltonian
ˆH =
i=1
ˆp2
i
2m
+
i
VNi + Vee + Vnuc
ˆp2
i
2m
is the non-relativistic kinetic energy operator
in heavy atoms (HA) the inner shell electrons has a speed comparable
with speed of light.
the core electrons of HA show sizable relativistic effects:
mrel =
me
1 − ve/c2
“response properties" like NMR shift is very sensitive to this effect.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 2
Dirac Atomic Theory
the marriage the Quantum Theory and Theory of Relativity
Dirac labor alone on the problem for 3 years before publishing it in
1928, "The Quantum Theory of the Electron"
(https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1928.0023)
if we compare Hamiltonians of Schrodinger and Dirac Equation, we can
see that there’s an α and β terms, unknown terms in which Dirac
introduced to proceed.
ˆH = −
1
2
2
+ v(r)
ˆH = α · p + βc2
+ v(r)
these supposed introduction of α and β terms (4×4 matrices) naturally
suggest its four component nature as well as positive and negative
energy solutions.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 3
Relativistic Quantum Chemistry
relativistic corrections to non-relativistic energy value
HR
= HNR
−
p2
8c2
+
∆v
8c2
+
1
2c2
1
r
dv
dr
1 · s
− p2
8c2 , mass-velocity : variation of the mass with the velocity; affects
contraction and stabilization of s and p shells ; expansion and
destabilization of the d and f shells)
+ ∆v
8c2 , Darwin: correction potential energy
+ 1
2c2
1
r
dv
dr
1 · s spin-orbit coupling: interaction of the electron spin and
the orientation of electronic motions (orbital angular momentum)
accurately described by the “fine structure” splitting of the Hydrogen
spectrum (Michelson-Morley, 1887).
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 4
from full Dirac to ZORA equation
Two-Component Relativistic Methods
the high computational cost of four-components relativistic
calculations has motivated the development of
computationally less demanding two-component
Hamiltonians.
two component relativistic Hamiltonians (involving only
positive energy orbitals): pseudopotential and all- electron
methods.
ZORA: accurate and efficient relativistic DFT
the zeroth order regular approximation (ZORA) to the Dirac
equation accurately and efficiently treats relativistic effects
in chemistry and can be applied with scalar and spin-orbit
corrections.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 5
from Dirac to ZORA equation
from Four-component Dirac
Vnuc + Jφφ − Kφφ + Jχχ cσ · p − kφχ
cσ · p − kχφ −2c2 + Vnuc + Jφφ − Kφφ + Jχχ
·
φ
χ
= E
φ
χ
(after a long derivation) to Two-Component Zeroth Order Approximation
HZORA
ΨZORA
= EZORA
ΨZORA
HZORA
= V + p
c2
2c2 − v
p + p
c2
2c2 − v
σ( V × p)
the approx. can now isolate scalar and spin-orbit effects
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 6
ReSpect vs ADF
ReSpect (RelativisticSpectroscopy) program enables the
prediction and understanding of molecular and material
properties using full Dirac Hamiltonian.
(http://www.respectprogram.org/ )
ADF (Amsterdam Density Functional) program enables the
prediction and understanding of molecular and material
properties using Zeroth Order Approximation (ZORA)
(https://www.scm.com/product/adf/)
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 7
Activity 1: ADF
recall, the NMR Shifts [δ] of specific atom (i) in a molecule is obtained
by subtracting NMR shielding [σ] constants from a reference:
δi = σref − σi
ADF NMR Shielding calculations
calculate the NMR properties of hydrogen in HI, HCl, and Benzene
use the prepared input files distributed in IS ( HCl_Scalar.inp,
nmr.inp, etc)
take a look at the script run_adf.sh to see how adf job submission
works.
NOTE: there are two parts of adf calculation, SCF
module-calculation and the NMR shielding calculation.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 8
Activity 1: ADF, cont.
after the successful NMR run, check the .out file, look for the keyword
“PARAMAGNETIC (), DIAMAGNETIC () and SPIN ORBIT (NMR
SHIELDING TENSOR)” and below the “isotropic” shielding value(s)
of the atom(s) sought.
NOTE: PARA eigenvalues is associated with ground and excited state
transitions, DIA contribution is only associated with the ground-state
(this is beyond the scope of our session).
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 9
Activity 1: ADF, cont.
after the successful calculations, ZORA Scalar and ZORA Spin-Orbit
approximations, compare the values with the experiment, complete this
table:
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 10
Activity 2: ReSpect
ReSpect NMR Shielding Calculations
calculate the NMR properties of hydrogen in HI, and HCl
use the prepared input files distributed in IS
take a look at the script run_respect.sh to see how ResPect run
works.
NOTE: analogous to ADF, there are two parts of ReSpect
calculation.
SCF module-calculation of unperturbed ground state MO
coefficient, in our case we started a guess SCF from (ksdkh2/pbe0)
to (mdks/pbe0, i.e. Dirac Kohn Sham Implementation)
CS module-calculation of the chemical shift properties
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 11
Activity 2: ReSpect, (Cont.)
after the successful CS calculation, check the .out_cs file and look for
the keyword “CHEMICAL SHIELDING FOR NUCLEUS (your ATOM)”
and below is the “Principal values of the NMR shielding tensor” where
the PARA and DIA part shielding value(s) of the atom(s) sought.
NOTE: in this full Diract calculation, the “SO effect" is embedded in
PARA and DIA, thus we can’t isolate the SO effects as opposed to the
ADF ZORA calculations.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 12
Activity 1: ResSpect, cont.
after the successful calculation, compare the values with the
experiment, complete this table:
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 13
Questions
which system(s) are sensitive to relativistic effects?
from these activities, which method has the calculated value(s) that
corroborates best with experimental 1
H NMR shifts?
is it worth choosing a computational method that contains
approximations that are cheap and has a practical results?
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 14
END
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 15
ANSWER KEY
The SO effect as we can see here becomes more significant as the
system involves heavier atoms (HCl to HI). The full Dirac calculation
provides the best results based on these geometries.
(Prepared by Radek Marek Research Group)
Lesson 11 - Relativistic Calculations 16