Download J - Laboratory of Molecular Interactions

Quantum chemical modelling of
molecular properties - NMR
shielding constant, spin-spin
coupling constant, rotational g
factor, spin-rotation constant,
nuclear quadrupole coupling
constant
Effective NMR Hamiltonian
High resolution NMR spectra are interpreted by means of effective
NMR Hamiltonian. Its eigenfunctions correspond to the spin states
of nuclei, and its eigenvalues to the energy levels of nuclear spins
in external magnetic field:
H
NMR
=−
X
K
1 X T
mK (DKL + KKL ) mL
B (1 − σ K ) mK +
2
T
K6=L
σ K is the shielding constant tensor, KKL is the tensor of (reduced)
spin-spin coupling constant (indirect, through-electron coupling).
DKL is the tensor of direct (through-space) spin-spin coupling
constant. All effects connected with the electronic structure are
contained in σ K and KKL .
(In some cases the contribution containing the nuclear quadrupole
coupling constant is added, but in isotropic medium it influences
only the width of the sygnal.)
The shielding constant can be treated as modification of external
magnetic field by the field resulting from the presence of moving
electrons.
Bloc
K = (1 − σ K ) B
Effective NMR Hamiltonian in isotropic medium
When molecules can rotate freely (gas, liquid), the effective NMR
Hamiltonian has the form:
NMR
Hiso
=−
X
B (1 − σK ) mK,Z
K
1 X
+
KKL mT
K · mL
2
K6=L
σK (isotropic shielding constants) i KKL (isotropic reduced
coupling constant) are traces of the appropriate tensors:
σK
=
KKL
=
1
Tr σ K
3
1
Tr KKL
3
Tensor DKL is traceless and averages out to zero.
σK is connected with the chemical shield δK
δK =
σK (reference) − σK (sample) ∼
= σK (reference) − σK (sample)
1 − σK (reference)
Shielding constant as energy derivative
Nuclear shielding constant can be expressed as a second
derivative of the energy with respect to external magnetic field and
nuclear magnetic moment
2
∂ E(B, mK ) ∂B ∂mK = σK − 1
B=0,mK =0
For closed shell systems (in nonrelativistic theory) shielding
constant (similarly as magnetic susceptibility) can be expressed as
a sum of diamagnetic part (calculated as an expectation value) and
the paramagnetic part (calculated as linear response function):
K(PSO)
B,K
σK,αβ = h0|Ĥαβ
|0i + hhm̂α ; Ĥβ
where
para
dia
ii = σK,αβ
+ σK,αβ
B,K
Ĥαβ
d2 Ĥ
=
dBα dBβ
and
K(PSO)
Ĥβ
=
dĤ
dBβ
Shielding constant as energy derivative, continued
σK , like any other second derivative of energy with respect to
time-independent perturbations, can also be calculated from
sum-over-states expression:
σK
E
D B,K = 0 Ĥ
0 + 2
ED E
D B,orb K(PSO) n n Ĥ
0
X 0 Ĥ
n6=0
En − E0
Electronic Hamiltonian in magnetic field
Standard Schrödinger Hamiltonian does not contain any terms
containing magnetic moments of the nuclei (or external magnetic
field). They appear in the Breit-Pauli Hamiltonian, which can be
2
order
derived when relativistic effects are taken into account to αfs
(αfs = c−1 is a fine structure constant).
In the presence of magnetic perturbations, the non-relativistic
molecular electronic Hamiltonian may in atomic units be written in
the form
X ZK
1X 1
1X 2 X
tot
mi · B (ri ) −
π −
+
H (B, M) =
2 i i
r
2
rij
iK
i
iK
+
i6=j
X
1 X ZK ZL X
tot
−
MK · B (RK ) +
MTK DKL ML
2
RKL
K6=L
K
K>L
We have here introduced the operators for the mechanical
momentum
π i = −i∇i + Atot (ri )
(1)
where Atot (ri ) is the vector potential at the position of electron i,
constructed such that its curl reproduces the magnetic induction
Btot (ri ) arising from the external field and the NMR active nuclei:
Btot (ri ) = ∇i × Atot (ri )
The vector potential and the magnetic induction may each be
decomposed into one contribution from the external field and one
contribution from each nucleus
X
tot
AK (ri )
A (ri ) = AO (ri ) +
K
B
tot
(ri )
=
B+
X
BK (ri )
K
The vector potential associated with the nucleus may for example
be written as
2 MK × riK
(2)
AK (ri ) = αfs
3
riK
andthe corresponding magnetic induction (the curl of AK ) as
BK (ri ) =
2
2 riK 1
−αfs
2
8παfs
− 3riK rTiK
δ (riK ) MK
MK +
5
riK
3
Ĥ B,K i Ĥ K(PSO) operators contributing to the shielding
constant
Consequently, Ĥ B,K i Ĥ K(PSO) operators (derivatives of the
Hamiltonian) have the following form:
Ĥ
B,K
=
2
αfs
Ĥ
e2 X (riO · riK ) 1 − riO rT
iK
3
2me i
riK
K(PSO)
=
2 2µB
αfs
X liK
3
~ i riK
Dependence of the calculated shielding constant on the
gauge origin
Both dia- and paramagnetic part of the shielding constant are
gauge-dependent (they change their values when the gauge origin
for the vector potential A is shifted) even for the exact wave
function and therefore separation of the shielding into dia- and
paramagnetic parts is somewhat arbitrary. As a consequence, their
separate discussion makes sense only when the gauge origin is
defined. The usual way is to put it on the nucleus of interest, and in
para
this case σK
is connected with the nuclear spin-rotation constant
(but the relation holds only in nonrelativistic approximation).
The total shielding constant depends on the gauge origin for
approximate wave function (except for wariational wave functions in
the limit of complete basis set). This dependence can be
eliminated by the use of London atomic orbitals (GIAOs):
i
ωµ (B, RM ) = exp − eAM · r χµ (RM )
~
However, the division into dia- and paramagnetic parts is still
gauge-dependent.
Computational requirements of the shielding constants
• Perturbation-dependent basis sets (London orbitals) are
absolutely necessary.
• A basis set should have a good description of outer core-inner
valance shell, and also polarization functions.
• The role od electron correlation varies with the system and
nucleus. The electron correlation effects tend to be large in
systems with multiple bonds. They are small in the case of 1 H
shielding, but large for example for 17 O shielding. This
phemonenon is connected with the relative magnitude of diaand paramagnetic parts: 1 H shielding is dominated by the
diamagnetic part, which is less sensitive than the
paramagnetic part to electron correlation.
• Shielding constants are strongly affected by rovibrational
effects and environment (including solvent). The role of
relativity is also significant (starting from 31 P shielding).
Spin-spin coupling constant as energy derivative
The spin-spin coupling constant between nuclei K and L can be
expressed as the second derivative of energy with respect to
nuclear magnetic moments MK and ML .
2
KKL + DKL
d E (M1 , M2 , · · · ) =
dMK dML
,
MK =ML =···=0
The reduced coupling constants KKL is connected with the
coupling constant JKL (measured in Hz) in the following manner:
JKL = h
γK γL
KKL ,
2π 2π
In the isotropic phases the trace of the tensor is measured:
JKL = 31 Tr KKL .
DKL is a traceless tensor.
DKL
µ0 3RKL RKL − 1R2KL
=
5
4π
RKL
Spin-spin coupling constant as energy derivative, continued
KKL , like any other second derivative of energy with respect to
time-independent perturbations, can be calculated from
sum-over-states expression:
KKL
= 0 2
∂ H 0 +2
∂MK ∂ML ED E
D ∂H ∂H X 0 ∂MK n n ∂ML 0
n6=0
En − E0
After evaluation of the derivatives (from the Breit-Pauli Hamiltonian
with no relativistic terms)
Ramsey formula is obtained
KKL
DSO = 0 hKL 0 +
+
PSO PSO T 0 hK nS nS (hL ) 0
2
EnS − E0
nS 6=0
FC T
SD SD T X 0 hFC
nT (hL ) + (hL ) 0
K + hK nT
.
2
EnT − E0
n
X
T
The first summation is over excited singlet |nS i states with
energies EnS , the second is over excited triplet |nT i states with
energies EnT .
The spin-spin coupling constant is calculated in practice as a sum
of an expectation value h0|Ĥ K,L(DSO) |0i (diamagnetic spin-orbit
term, usually small) and terms calculated as linear response
functions: hh Ĥ K(PSO) ; Ĥ L(PSO) ii (paramagnetic spin-orbit term) i
hh Ĥ K(FC) + Ĥ K(SD) ; Ĥ L(FC) + Ĥ L(SD) ii (Fermi contact term and
spin-dipole term).
hDSO
KL
=
hPSO
K
=
hFC
K
=
hSD
K
=
e2 X (riK · riL ) 1 − riK rT
iL
,
3
3
2me i
riK riL
X liK
2 2µB
αfs
,
3
~ i riK
X
2 8π
δ (riK ) 1,
−αfs
3 i
4
αfs
2
−αfs
X 3 riK rT − r 2 1
iK
iK
,
5
riK
i
α ≈ 1/137 is a fine structure constant, riK is a position of the
electron i with respect to the nucleus K, I3 is a 3:3 unit matrix,
P
δ(riK ) is a Dirac delta, and si is spin of electron i. i denotes
summation over all electrons, and T transpose of a vector.
If only the isotropic part of the spin-spin coupling constant is of
interest (which is usually the case), the FC-SD term does not need
to be calculated, since it is purely anisotropic and does not
contribute to isotropic JKL . JKL is then a sum of four terms:
DSO
PSO
SD
FC
JKL = JKL
+ JKL
+ JKL
+ JKL
Computational requirements of the spin-spin coupling.
• Calculations of spin-spin coupling constants are relatively
time-consuming: each magnetic nucleus introduces a new
perturbation(3 components).
• Electron correlation is crucial: Hartree-Fock produces
inaccurate or nonsensical results.
• DFT/B3LYP is suitable for most of the calculations — except for
fluorine coupling constants.
• Very accurate calculations for small molecules can be carried
out by means of MCSCF i CC. (In such calculations also
rovibrational effects should be accounted for.)
• Standard basis sets of atomic orbitals are not suitable for
calculations of spin-spin coupling constants: the FC term
requires a large number of s funcitons with high exponents
(inner core). The PSO and SD contributions, on the other
hand, need outer core, valance and polarization functions
(PSO is similar in this respect to the shielding constant).
Different contributions have thus different requirements.
What do we mean by ’inaccurate’ and ’nonsensical’ results?
C2 H6
C2 H4
SCF
Jeq
exp
Jtot
1
JCC
60.6
34.5
1
JCH
149.1
125.2
1
JCC
1270.2
67.6
1
JCH
754.5
156.3
T. Helgaker i M. Pecul Spin–spin coupling constants with HF and DFT
methods, in: Quantum Chemical Calculation of Magnetic Resonance
Properties. Theory and Applications.
Problems with DFT in calculations of 19 F coupling constants
HF
Jeq
B3LYP
Jeq
MCSCF
Jeq
CCSD
Jeq
exp
Jtot
HF
1
JHF
668.9
416.6
544.2
521.6
500
FHF−
2
JFF
656.8
24.9
358.1
438.6
≈ 247
1
JHF
181.0
38.4
126.5
81.6
107.0
T. Helgaker i M. Pecul Spin–spin coupling constants with HF and DFT
methods, in: Quantum Chemical Calculation of Magnetic Resonance
Properties. Theory and Applications, ed by: M. Kaupp and M. Bühl and
V. G. Malkin, Wiley-VCH (2004), pages 101—121.
Examples of
calculations.
Spin-spin coupling constants in C2 H2
1
a
J(CC)
1
J(CH)
2
J(CH)
3
J(HH)
MCSCF-3/cc-pCVQZsu2
184.68
244.27
53.08
10.80
Eksperymenta
184.52
242.40
53.76
10.11
with vibrational corrections
M. Jaszuński i K. Ruud Spin-spin coupling constants in C2 H2 , Chem.
Phys. Lett. 336 (2001) 473.
All spin-spin coupling constants (7587)in valiomycine C54 H90 N6 O18
(LDA/6-31G calculations).
M. A. Watson, P. Sałek, P. Macak, M. Jaszuński, T. Helgaker,
Chem. Eur. J. (2004) 10: 4627–4639.
62 Hz
77 Hz
Coupling constants in C60 fullerene.
M. Jaszuński, T. Helgaker, and K. Ruud Mol. Phys. (2003) 101:
1997—2002.
Vibrational effects on spin-spin coupling constants [Hz]
B3LYP
Jvib
exp
Jtot
HF
1
JHF
−38.0
500
H2 O
1
JOH
5.4
−80.6
CH4
1
JCH
5.3
125.3
2
JHH
−0.7
−12.8
1
JCC
−10.0
174.8
1
JCH
4.6
247.6
2
JCH
−3.0
50.1
3
JHH
−0.1
9.6
C2 H2
T. Helgaker and M. Pecul Spin–spin coupling constants with HF and DFT
methods, in: Quantum Chemical Calculation of Magnetic Resonance
Properties. Theory and Applications, ed by: M. Kaupp and M. Bühl and
V. G. Malkin, Wiley-VCH (2004), pages 101—121.
Nuclear spin-rotation constant
Interaction of magnetic moment of a nucleus with magnetic field
caused by molecular rotation results in splitting of the rotational
spectrum.
∆EK = −IT
K MK J
where IK is nuclear spin, J is a total molecular momentum, and
nucl
MK = Mel
K + MK
is a spin-rotation tensor. It is composed of two terms, nuclear and
electronic, which can be expressed as:
Mnucl
K
=
2
αfs
µ N gK
X
L6=K
i
2
RLK
1 − RLK RT
LK −1
I
ZL
3
RLK
Mel
K
µN gK ∂ 2 E el =−
2π ∂mK ∂J mK ,J=0
−1
Mel
hhĤ K(PSO) ; ˆlK ii
K can be obtained as a linear response I
Connection with the shielding constant
Mel
K is connected with the paramagneti part of the shielding
constant, defined as a difference between the total shielding
constant and its diamagnetic part with the gauge for the vector
potential A on nucleus K.
nucl
Mel
K = MK − MK
i
~ µ N gK h
−1
=
σ K − σ dia
K (RK ) I
µB 2π
Molecular magnetogiric ratio g
molecular g factor
A rotating molecule produces a magnetic field because of the
motion of charged particles: electrons and nuclei. The associated
magnetic moment can be expressed as:
mJa
µN X J
gab Jb
=
~
a,b
gdzie µN is nuclear magneton, Jb is the rotation quantum number,
and the molecular factor g J is a sum of contributions from nuclei
and electrons.
J
g =g
J,nucl
+g
J,el
=
−1
mp I
X
ZK
2
RK,CM
1
K
−
4mp me −1 para
I ξ
(RCM )
2
e
−
RK,CM RT
K,CM
I is the momentum of inertia, and ξ para is a paramagnetic part of
magnetic susceptibility (discussed in one of the previous lectures).
g influences high-resolution rotational specra even in the absence
of external magnetic field. It is important for precise determination
of molecular geometry.
Nuclear quadrupole coupling constant
Relaxation time in NMR (influencing the signal width) is strongly
influenced, for nuclei with spin 1 or higher, by the interaction of
nuclear electric quadrupole moment with electric field gradient:
∆ν1/2 ∝
(−
eQ
K
V
z
z
~
K
eQ
Vz z K
~
2
is the nuclear quadrupole coupling constant, NQCC,
Vz z electric field gradient on nucleus K, eQ nuclear electric
quadrupole moment. For nuclei with 1/2 spin (for which eQ = 0)
electric field gradient does not affect the signal width.
Electric field gradient on the nucleus influences also the fine
structure of microwave spectrum (which allows for determination of
nuclear quadrupole moments).
Nuclear quadrupole coupling constant, continued Electric field
gradient on the nucleus K, Vα β K , and therefore the nuclear
quadrupole coupling constant, can be calculated as the
expectation value:
K
K
Vαβ
= h0 | V̂αβ
| 0i
where the operator V̂αβ has the form:
K
Vˆαβ
=
2
1 X 3r̂iK,α r̂iK,β − r̂iK
δαβ
e
5
4πǫ0 i
riK
−
2
X
3RLK,α RLK,β − RLK
δαβ
1
e
ZL
5
4πǫ0
RLK
L6=K
Computational requirements of nuclear quadrupole coupling
constant
Operator in the expression for nuclear quadrupole coupling
constant is of r −3 type (like those in the expressions for the
shielding constant and PSO contribution to the coupling constant),
therefore the basis set requirements (extended outer core-inner
valance region) and the electron correlation effects are similar.
Programs for calculations of NMR parameters
• Shielding constants in Gaussian 09 (HF, DFT, MP2), Dalton
(HF, DFT, MCSCF, SOPPA), CFOUR (CC), ReSpect (DFT),
ADF (DFT), NWChem (HF, DFT).
• Molecular g factors in Dalton (HF, DFT, MCSCF, SOPPA).
• Spin-spin coupling constants in Gaussian 09 (HF, DFT), Dalton
(HF, DFT, MCSCF, SOPPA), CFOUR (CC), ReSpect (DFT),
ADF (DFT), NWChem (HF, DFT - in developement).
• Nuclear quadrupole coupling constants in Gaussian 09 (HF,
DFT), Dalton (HF, DFT, MCSCF, SOPPA), ADF (DFT), CFOUR
(CC)...