Giao Igai Cs
Giao Igai Cs
Giao Igai Cs
z
= m
I
where is nuclear spin function, nd m
I
will look like: m
I
= I, I-1, I-2, - I
Nuclear magnetic moment
N
is proportional to nucleus spin moment I
N
= g
N
N
I
= g
N
N
I
where
N
- nuclear magneton
c 2M
e
N
N
h
=
In magnetic field H
N
m
I
H
0.
Nucleus energy-level splitting in magnetic field subject to m
I
value can be measured with NMR spectroscopy technique; in doing
so the absorption resonant frequency
N
(Larmors precession frequency) is defined by the following expression:
h
N
= g
N
N
H
0.
Should the magnetic nucleus be located in atom or molecule, the local magnetic field in the area
of nucleus would differ from external magnetic field
0
due to magnetic nucleus being shielded by
electrons. The magnetic nucleus is enclosed in electron shell, in which magnetic field H induces the
current that in turn generates the secondary magnetic field H
local
. Thus, the condition of magnetic
resonance should undergo a change, and the absorption resonant frequency of alternating
electromagnetic field will differ for nucleus in substance and for "bare" nucleus. The nature of this shift
is double. Firstly, in the presence of magnetic field H the Larmors precession of electronic charges
around it is the equivalent of electric currents producing, in the area of nucleus, the secondary magnetic
field H
local
, which is added and is proportional to it. Secondly, external magnetic field
0
polarizes electron shells, the distortion of which
create, in the area of nucleus, magnetic field H
pol
, likewise proportional to
0
. The total field effecting the nucleus is:
H
local
= H
0
+ H
ind
+ H
pol
0 0
local
H H H = .
As a result, the condition of magnetic resonance for the nucleus, defined by equation h
N
= g
N
N
H
local
, will undergo changes on
account of nucleus shielding by electrons.
The degree of shielding is defined by nuclear magnetic shielding tensor .
When a molecule moves fast (as in the case of liquids), the average value of magnetic shielding tensor components is observed.
Tr
3
1
3
1
) (1 H
0
local
= =
= H
Energy W `, caused by magnetic shielding, will look like:
W` = H .
The electron behaviour in magnetic field H is described by replacing pulse with + (/) in Hamiltonian function
A P P
c
e
+
Here magnetic vector potential defined by the equations: divA = 0; rot A = H.
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 13
Magnetic dipole at a point with radius - vector r
k
generates the magnetic field having magnetic vector potential:
| |
|
|
.
|
\
|
=
=
k
3
k
k
k
r
rot
r
r
A
.
The vector potential, produced by magnetic field at a point of the electron location, has
the appearance:
In view of these potentials for the molecule containing n electrons and N magnetic nuclei, the Hamiltonian in the presence of
external magnetic field H (at non-relativistic electron description) has the following appearance:
A S A P rot 2 )
c
e
(
2m
1
U H
2
+ + =
= = = =
+ +
|
|
.
|
\
|
+ + =
n
1 k
N
1 q
q
k k
n
1 k
H
k k
N
1 q
q
k
H
k k
rotA S 2 A rot S 2 A
c
e
A
c
e
P
2m
1
U H
where U is electrostatic energy of the system. Here we leave out the magnetic interactions between electrons and
Zeemans interactions of nuclei with applied field, as well as their dipole interactions, for they bear no relation to the
problem under investigation.
Should the Hamiltonian, allowing for all magnetic interactions in molecule, is obtained for a system, it is possible to write the
expression for the energy of system, which would include the terms of the first and the second order with respect to the distortions -
magnetic field, nuclear magnetic moment
...
,
,
H
2
1
0
W W + + =
.
Since the interaction energy of nuclear dipole with electron shell is low, it can be calculated by means of the perturbation method.
The indirect spin-spin interactions, other than usual dipole interactions, may arise at the expense of electrons as follows. Nuclear moment
N1
generates a field that distorts electron shells. The electron shells so distorted build up magnetic field
, proportional to
N1
value
,
in
the area of other nuclear moment
N2
. Interaction of this field with
N2
causes bilinear, relative to I
N
and I
N
, interaction. As a result, spin-
spin interactions between I
N
and I
N
spins are controlled by the energy decomposition terms, bilinear with respect to these spins.
One may single out three contributions to spin-spin interaction constant:
N N N N N N N N N N
J h J h W I I I I
+ =
N N N N N N
hJ W
= I I
(
+ + =
+ + =
k
2
kN
N
3
kN
kN
N N k 1
3 2 1
2
1
r
c
e
i 2m
1
H
H H H H
r H
r
I h
h
H
1
describes the interaction of nuclear spin with electron orbital motion;
( )( ) ( ) | |
=
k N
3
kN N k
5
kN kN N kN k N 2
r r 3 2 H I S r I r S h
2
describes dipole-dipole interactions;
=
k N
N k kN N 3
) (
3
16
H I S r
h
3
describes contact contribution.
There are a number of definitions of magnetic shielding tensor, which result in different ways of calculating the tensor components of
nuclear magnetic shielding. According to one of them, the tensor of magnetic shielding is formally defined as the second order term in the
decomposition of total molecular energy, proportional to the product of nuclear magnetic moment and magnetic field components, hence
magnetic shielding tensor is defined by the second order derivative of molecular energy on H and magnetic moment .
( )
,
K
E
H
H o
=
|
\
|
.
|
|
=
2
= + +
1
3
( )
xx yy zz
Magnetic susceptibility is found in the same way
| |
k
H
k
2
1
A r H =
Full Paper _______________________________________________________________________________________ R.M. Aminova
14 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
=
2
E
H H
.
Magnetic shielding and magnetic susceptibility is the second order effects with respect to the result of weak interactions in
molecular system. Compared to Coulombs interaction energy and kinetic energy (having the order of a few electron volts (eV)), the
magnetic effects make up 10
-4
-10
-14
eV. This is one of the difficulties hampering their evaluation, since the necessity arises for the
disturbance effects of the second order to be allowed for with strict precision.
Along with defining the shielding tensor as the second mixed derivative of the energy of electronic system (definition (1)),
simultaneously disturbed by uniform magnetic field and non-uniform field of dot magnetic dipole (nuclear spin moment), the component
of nuclear magnetic shielding tensor can also be calculated as follows [26, 27]:
Definition (2) - - is the component of the secondary magnetic field H
parallel to axis .
Definition (3) - - is the component of magnetic moment generated at the molecules electron shell owing to its being disturbed
by non-uniform magnetic field of the nucleus dot magnetic dipole moment aligned with axis .
While in theory all three ways of calculation are equivalent, in actual practice the first way is most commonly used, though the
second one is the best suited as far as it is connected with the most simple magnetic disturbance, arising in calculating the current induced
in electron shell of a molecule by uniform magnetic field, which can be calculated by the following well-known formula:
The starting point for the development of quantum-mechanical calculation methods of magnetic shielding tensor has been
Ramsey's classical work ([4], see also [5]), in which the expression for the components of nuclear magnetic shielding tensor was obtained
on the basis of definition (1) and conventional disturbance theory
+ =
=
n k k
k
k
n n k n
p
k
k k k k
d
c c d
r
M
d M E E
d r r r r
mc
e
. . ) ( 2
) / ) ( (
2
0 3 0
1
0
0
3 2
0 2
2
In the course of two decades these Ramsey's equations were used as the basis for many approximate methods describing
chemical shifts, semiempirical as well as non-empirical (for example, see [6, 7]).
2. Variational methods.
Along with Ramsys perturbation theory, some traditional variational methods [8-15] have been developed in order to calculate
the second order magnetic properties. One of their major advantages is that they do not require the knowledge of continuous energy
spectra of the molecule, which is of great importance. In following the variational method, a full wave function of molecular system can
be broken down into a series in which the linear, with respect to magnetic field, correction to wave function will define the magnetically
perturbed wave function of the molecule. The latter is written as the product of unperturbed wave function
multiplied by some
correction function g(r) of electron coordinate.
Here g is the variational function that allows for magnetic polarization of electronic distribution in molecular system and may be
presented as a[8-15] polynom dependent on electron coordinate.
Then, the expression for paramagnetic contribution to shielding can be obtained with definition (2), as - component of the
secondary magnetic field H
generated by the current at nucleus, which is induced by homogeneous magnetic field , parallel to axis ,
in the electronic shell. On calculating the value of current by the formula
,
we find the magnetic field generated by this current by Bio-Savar-Laplace formula
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 15
In works [13-15] the formalism of variational perturbation theory was considered. This approach was proved to successfully
predict the chemical shifts of diatomic molecules both for light and for heavy atoms. The method was expanded further in Sadleys
works [16-19]. In particular, the coupled variant of this method was described.
It was not until the second part of the 80th, when the computer technologies came to extensive use, that the realization of most
correct quantum-chemical methods and approaches became possible. It was also favored by the creation of novel, mathematical methods
and the algorithms that allowed to enhance the processes of optimization, methods of analytical calculation for derivatives and integrals.
So, for example, GIAO method, proposed by Dichfild in 1974 [32], had remained unused for 12 years, until the method and algorithm of
analytical calculation for derivatives was developed by Wolinscy and Pulay [33, 34]. This has permitted for the chemical shift
calculations to be substantially quickened.
3. Calibration invariancy problem.
One of the basic problems inherent in calculation of nuclear magnetic shielding tensor is the calibration invariancy. Among
others two more problems are worth to be singled out: the selection of basis and the allowance for electron correlation effects.
We would like to discuss the first problem in greater detail since all the recent methods proposed in literature include, in one
form or another, various calibrations (from zero) of magnetic vector potential.
In its simplest form the calibration problem is a nonphysical dependence of calculation results on arbitrary origin of magnetic
vector potential.
It is worth emphasizing that at exact calculations both diamagnetic and paramagnetic contributions depend on the calibration of
magnetic vector potential, while the shielding constant itself should not be dependent on the selection of magnetic vector potential origin.
It is common knowledge that magnetic field H enters into Hamiltonian not directly, but via vector potential A, which is defined
with a precision of arbitrary scalar function. A gradient of any arbitrary function f may be added to vector potential.
A --->
= A + f.
In this case the equations of electromagnetic field (H = rot A) do not vary, as rot(f) =0
As for uniform magnetic field, the simplest form of vector potential is = [ r].
When the origin of coordinate system shifts into new origin R so that r
k
= r
k
+ R (here r
k
- radius - vector of electron in former
co-ordinate with the origin at point , and r
k
- radius - vector to the electron in new system of coordinates with the origin at point
, R
- radius - vector from point O to point
+ +
+ +
=
E
n
-E
0
excitation energy.
Green's function being known, it may in principle give the information about excitation electronic energies and oscillatory forces,
which can be used for calculating many observable values, such as polarizability, constants of spin-spin interaction and magnetic
shielding.
The motion equation in energy sense has the appearance:
| | | |
E E
H Q, P; 0 Q P, 0 Q P; E + =
.
The motion equations within the framework of such formalism can be solved at different approximation levels, depending on the
order of fluctuation potential decomposition (fluctuation potential is electron repulsion minus Fock potential). In zero approximation,
this formalism is adequate to uncoupled variant of Hartley-Fock method. The first order approximation of polarization propagator is
designated as Random-Phase Approximation (RPA). RPA and Hartley-Fock methods are equivalent for static properties (Coupled
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 17
Hartree-Fock - (CHF)). The second order approximation of polarization propagator is named SOPPA.
In this formalism magnetic shielding tensor has the appearance:
0 E
c
j
3
kj
kj
2 2
2
P
k
j
j cj c c j cj
'
n
0 n
c
3
kj kj
2 2
2
P
l ;
r
l
c 2m
e
, r i l , R r r
E E
c.c. 0 l n n /r l 0
c 2m
e
=
= =
+
=
h
Thus, the effects of electronic correlation for shielding paramagnetic contribution are allowed for in SOPPA method to some
extent. Diamagnetic contributions are not sensitive to electronic correlation effects (see [43-45, 46]). It should be noted that this method
gives gage-invariant results providing that at each atom, even in two-nuclear molecule, there are no less than three d-functions!
The condensed table 1, given below, sums up the basic methods which have been developed in recent years.
Table 1. Ab initio methods of calculation of Nuclear Shielding.
CHF (Coupled Hartree-Fock SCF perturbation theory)
(Stevens, Pitzer, Lipscomb, 1963)
CHF (using very large and flexible basis sets): Holler, Lischka, 1980;
Lazzeretti, Zanasi, 1980
CHF + GIAO (Coupled HF SCF perturbation theory using of gauge-invariant
(gauge -including) atomic orbitals): Ditchfield, 1974
The use of GIAO has become widespread.only since the presentation of an efficient
implementation of the approach using modern analytical chemical shift calculations derivative
techniques by Wolinski, Hinton, Pulay (1990)
IGLO (Individual Gauge for Localized Orbital): Kutzelnigg, 1980
LORG (Localized Orbitals/ Localized Origin): Hansen, Boumann, 1985
This method was formulated within the random-phase approximation (RPA)
SOLO LORG (RPA) + SOPPA (second-order polarization propagator approximation):
Bouman, Hansen, 1990
IGAIM: Keith, Bader have introduced the Individual Gauges for Atoms in Molecules, 1992
CSGT: Keith, Bader have developed novel approaches for computing molecular magnetic properties via accurate determination of the three-dimensional electronic
current density induced by external uniform magnetic fields, Continuous Set of Gauge Transformations method , 1993
(Rebane T.K. in 1960 offered the idea of variating of magnetic vector potential!)
The methods of shielding tensor calculation on the basis of density functional theory have come under development relatively not
long ago. They were started within the framework of
+ +
+ + =
Here (1) - kinetic energy of noninteracting system with density (r), E
xc
- so-called exchange-correlation energy.
Application of variational principle for E
el
total energy expression leads to Kohn-Sham equations:
(r) (r) (r) V dr
r r
) (r
V(r)
2
1
i i i i xc
'
'
'
2
=
|
|
.
|
\
|
+
+ +
Here
| |
(r)
(r) E
(r) V
xc
xc
=
.
If E
xc
is not taken into account, physical content of the theory becomes identical to Hartley's approximation.
Hohenberg-Kohn's theorem has been proved not to be valid in the presence of magnetic field. To solve this problem the density
functional theory, with current density vector included, had to be developed. CDFT (Current density functional theory) (C.van Wllen,
1994), in which exchange-correlation energy (exchange-correlation density functional) is the functional of both density , and
paramagnetic current density J
p
(r):
= 2
i
*
i
J = -i (
k
*
k
-
k
k
*).
Full Paper _______________________________________________________________________________________ R.M. Aminova
18 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
H
C
O
O
H
H
C
O
O
H
H
C
O
O
H
H C
O
O
H
H C
O
O
H
H
C
O
O
H
H
C
O
O
H
H
C
O
O
H
H
C
O
O
H
( I )
( I I )
( I I I )
It should be emphasized that in this case J is not a gradient-invariant value. Vingale and Rasolt [111, 112] introduced novel,
gradient-invariant variable : =x(J/) and showed that any exchange-correlation functional, allowing gage-invariant theory, can be
expressed as the functional of and .
In the framework of CDFT and using both IGLO, and GIAO approximations, C. van Wullen [71] presented such equations for
CS calculation, with the help of which the problem of calibration origin has been solved.
Summing up the results of the carried out analysis, we can state that GIAO method in combination with CHF (Coupled Hartree-
Fock) formalism is now undoubtedly most attractive for shielding constant calculations. These orbitals are physically admissible as they
make up the exact solution in magnetic field for one-electron one-center problem. It immediately follows that the basic decomposition
has to be shorter. There are no problems with calibration selection, which usually hampers calculations with finite size bases, as far as
GIAO is not dependent on the origin of magnetic vector potential. It means that the magnetic properties can be immediately calculated
for any, correlated or noncorrelated, wave function, irrespective of calibration invariancy problem.
6. Calculations of nuclear magnetic shielding constants in complexes with hydrogen bond.
Using GIAO method, we have calculated the structure and constants of nuclear magnetic shielding for a number of associates of
different structure formic and acrylic acids, as well as their complexes with solvents - water and dimethylsulfoxide [72].
The analysis and revelation of potentiality of current nonempirical methods have been of certain interest for interpreting
chemical shifts in different structure complexes with hydrogen bonds and for studying the interrelation between the structure, electronic
distribution, complex geometry and constants of nuclear magnetic shielding.
All calculations have been carried out by ab initio method at RHF-SCF level of the theory with basic sets 6-31G*, 3-21G*.
Influence of electronic correlation was allowed for by Moller-Plesetta method in the second order of perturbation theory (MP2), as well
as by DFT method [73, 74] on the basis of three-parameter hybrid Beckes method [75] using Lee, Yang and Parr correlation functional
(B3LYP) that includes local and nonlocal terms. On the one hand, the calculations have allowed the establishment of the associate spatial
structure and hydrogen bond energies. So, as for trimeric and tetramerous forms of formic acid, our calculations have indicated a flat
complex structure (fig. 1, II), and the attempts to construct a bicyclic structure by means of the appropriate calculations with optimizing
geometrical parameters have not been successful. Within the framework of one method the correlation between hydrogen bond length
and its energy is observed, namely, the greater H-bond length, the smaller is hydrogen bond energy.
According to numerous experimental data, the association of molecules results in great shift of NMR signals of hydrogen-bonded
proton into weak fields.
The calculation data as a whole confirms the experimentally observable tendencies. Data analysis for monomer, dimer, trimer
and tetramer shows that CS of acid group protons of all associates is bound to shift into weak fields. Comparing the data for dimer and
monomer indicates the formation of hydrogen bond r(O..... H*) will shift acid proton signal into weak fields up to 6 ppm (according to
the results of B3LYP/6-31G // B3LYP/6-31G method). This value is in agreement with experimental data. So, according to Pople data
[76], the difference in CS of monomer and dimer of acetic acid has the order = 5.65 ppm
The calculation results of shielding constants for light nuclei, such as protons, are slightly dependent on the method chosen and
are in conformity with available experimental data as a whole (see below formic acid water solution). At the same time, the calculated
values of magnetic shielding constants for oxygen atom in formic acid self-associates are rather critical to the basis selection.
In these systems
13
carbon signal of carbonyl group shifts into weak fields as well. Comparative analysis of shielding constant
changes for monomer and self-associates of formic acid (n=1-4) indicates that
13
magnetic shielding value in dimer decreases, as
compared to monomer, approximately by 9-10 ppm. There has been observed the correlation between hydrogen bond length and
shielding value of fixed hydrogen atom. There is a noticeable correlation between bond length and chemical shift - the shorter the bond,
the greater is the value of chemical shift into weak fields.
It is known that solvents can have an essential influence on chemical shifts.
We have performed the calculations of structure and constants of nuclear magnetic shielding for various associates of formic and
acrylic acids with such solvents as water and dimethyl sulfoxide. The structures of the designed complexes with solvents are shown for
formic acid in fig. 4, and for acrylic acids - in fig. 5.
It is generally considered that acetic acid in water solution exists in the form of cyclic complex containing two water and one
acid molecules. The calculations of complex formic acid-two water molecules have been performed by us (fig. 4). Optimization of all the
parameters results in
a cyclic structure of the complex presented in fig. (4, V). Hydrogen atoms of two water molecules, not involved in the formation of
hydrogen bonds with formic acid molecule, are located on different sides of the plane with molecule and - bonds of two
water molecules participating in hydrogen binding.
Fig. 3. The structure of dimer, trimer and tetramer of formic acid.
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 19
Table 2 shows the calculation results for atomic charges and magnetic shielding
constants. One can see that chemical shifts of oxygen, as in the case of formic acid self-
associates, essentially depend on the calculation technique for correlation corrections -
2 or B3LYP. In our opinion, one of the reasons for it is that both methods result in
noticeably different charge values, especially with oxygen atom. This fact exerts but a
little effect on the shielding of the hydrogen atom.
Fig. 5. Complexes of acrylic acid with water and dimethyl sulfoxide, investigated in this work.
To compare the calculated values of magnetic shielding with the experimental ones we will further turn to the experimental data
given in works [79-80]. According to these data, the chemical shift of
13
nucleus for formic acid in water is equal to 166 ppm.
Table 2. Values of atomic charges q and chemical shifts s for the solutions of formic acid in water, obtained by different calculation techniques.
1,2
Basis H
3*
O
1
-H O
4
= C
2
-H H
5
-C H
7*
O
6*
O
9*
H
10
H
8
H
11
MP2/6-31G*
E=-340.812283975
q 0.547 -0.717 -0.607 0.563 0.184 0.517 -0.952 -0.939 0.511 0.448 0.445
20.50 151.86 -19.89 32.76 24.51 25.94 312.64 315.86 27.43 31.06 31.10
B3LYP/6-31G*
E= -342.622953912
q 0.447 -0.544 -0.461 0.374 0.139 0.433 -0.816 -0.812 0.436 0.404 0.402
20.35 112.81 -23.02 35.80 24.39 26.10 296.00 301.21 27.60 31.49 31.46
1
Numbering of atoms is given in fig. 4.
2
Experimental value, equal to 19.5 ppm for , is obtained on the basis of CS data given in [5]. CS of formic acid proton in water solution (scaled in )
() (
2
) =-3.9 ppm of benzene. By assuming CS value for benzene protons scaled in to be equal 7.24 ppm [5], we obtain the value of 11.4
ppm Using 30.62 ppm for absolute magnetic shielding of proton in tetramethylsilane, we obtain 19.48 ppm for absolute shielding of formic acid proton in
water.
For comparison, experimental data for clear water are as follows: () =334 ppm, () = 30.05 ppm.
Hereinafter,
1
and
13
chemical shifts are given in scale, in which for tetramethylsilane (TMS) is equal to 0. The value of
absolute magnetic shielding for TMS proton [81] is taken to be equal 30.62 ppm. The value of absolute magnetic shielding for
13
nucleus in TMS is 195.05 ppm [30], in view of CS for
13
in methane molecule being equal - 2.3 ppm [80, 82], and absolute magnetic
shielding value for
13
in methane molecule is 197.35 ppm [83].
Taking 195.05 ppm to be the value of absolute magnetic shielding of
13
in tetramethylsilane, we obtain 29.05 ppm as the
absolute magnetic shielding value of
13
for formic acid water solution.
13
CS of carbonyl group in formic acid water solution makes up 27 ppm relative to CS
2
. CS for
13
in CS
2
is equal to 192.8
ppm ( scale ranging from TMS to low fields). Then, CS for
13
in makes up 165.8 ppm Taking 195.05 ppm as the value of
13
absolute shielding for TMS, we obtain 29.2 ppm as the value of
13
absolute shielding for formic acid water solution.
The calculated value of chemical shift for the proton involved in the formation of hydrogen bond between formic acid and two
water molecules agrees well with the experiment, as well as
13
CS of carbonyl group.
When hydrogen bond =......- is opened,
-13
signal of COO-group shifts into high magnetic field by 6.5 ppm [84]. In
calculating
13
chemical shift of monomer by 2/6-31G
*
method, the constant of
13
magnetic shielding is equal to 39.01, in
the complex with water ...2
2
the calculated value of
13
shielding constant makes up 32.76 ppm Thus, the difference is 6.25
ppm As for B3LYP/6-31G* method, in a monomer
13
chemical shift is equal to 41.77 ppm, in complex with water
13
CS in COO-
group is 35.80 ppm. In this case the difference is 5.97 ppm. Thus, the calculations correctly predict the tendencies of CS changes in
complexation.
Finally, we have performed calculations for the complex of formic acid with dimethylsulfoxide (DMSO). According to some
literary data, while water forms intermolecular hydrogen bonds (IHB) with the acid both by =, and --bonds, in the acid solution
with DMCO IHB with acid O-H-group alone is believed to be possible. Our calculations have revealed the existence of two forms of
complexes. In the first form, IHB is realized between oxygen atom of S=O-group and hydrogen atom of formic acid O-H-group, the
plane, in which acid molecule is located, being parallel to the base of pyramid consisting of -atoms of dimethylsulfoxide (fig. 4, VI).
Fig. 4. Dimer of formic acid and complex of
formic acid with water and dimethyl sulfoxide,
investigated in this work.
H
5
C
2
O
1
O
4
H
3
H
7
O
6
H
8
H
10
O
9
H
11
(IV)
(VI)
S2
C4
C3
O1 H
O C
O
H
H10
H6
H5
H7
H8
H9
1,475
2,20
2,20
(V)
C
1
S
2
C
6
O
7
H
4
H
8
H
10
H
3
H
9
H
13
O
11
C
12
H
15
O
14
H
5
1,618
1,618
C
C C
O
O H
H
H
H
C
C C
O
O H
H
H
H
15
14
13
12
11
9
8
7 6
5
4
3
2
1
2,15
2,08
2,03
C
C C
O
O
H
H
H
H O
H
H
H O
H
C
C C
O
H
H
H
H
(X)
8
7 4
6
3
2
1
1,49
2
,
1
9
2
,1
9
C
C
C
O
H
H O
H
O
S
C
C
H
H
H
H
H
H
10
(IX)
(VIII)
(VII)
Full Paper _______________________________________________________________________________________ R.M. Aminova
20 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
In the second form, more energy preferable, the complexation is seemingly observed in unexpected, cyclic orientation (fig. 4, V). In this
case the conventional, chain structure has substantially lower energy of complex formation (with RHF/6-31G
*
method total energy makes
up 740.284005021 .u.), as compared to the energy of the complex with two bond directions (total energy is 740.297391424 .u.). In
cyclic structure (fig. 4, V) there are two intermolecular hydrogen bonds - the first, as it should be, by sulfogroup - S=....-, and the
second - by carbonyl acid group, with capturing one hydrogen atom from methyl groups. In cyclic complex the plane of formic acid and
S=O bond are rigorously located in the bisector dividing the angle between S-C-bonds in half.
3
-groups are symmetric about this
plane. Hydrogen atom of OH-group in formic acid is coordinated to oxygen atom of S=O-group, and oxygen atom of carbonyl group
S=O is located at the same distance (2.2 A) from two hydrogen atoms (
4
and
8
), each belongs to one of dimethylsulfoxide methyl
groups. Other atoms of both methyl groups (
3
,
5
,
9
,
10
) are farther removed from oxygen atom of carbonyl group (their distance to
oxygen atom makes up 3.8 A). The conformity between the orientation of methyl group protons about S=O-group and their chemical
shifts is observed; CS for
4
and
8
protons are equal to 28.85 ppm, and for protons
3
,
5
,
9
,
10
- about 31 ppm Thus, the signals of
DMSO methyl group protons, involved in binding, have to be shifted into weak fields ( = 28.56 ppm), as compared to unbonded methyl
group, which can be indicative of association. In this case there is observed the correlation between CS and charge; the latter grows from
0.22 to 0.29 (electronic density is decreased) in comparison with isolated dimethylsulfoxide molecule. Hydrogen bond by S=O-group is
much shorter, with the result that the constant of acid proton magnetic shielding decreases and the signal from this proton shifts into
weak fields, with the constant of magnetic shielding of
13
carbonyl group decreasing as well. Another bond is longer, but, obviously, it
is its formation that increases, more than two time, the binding energy of the complex formic acid-dimethylsulfoxide. Extraordinary
strong binding is also indicated by rather short length of ..(S)... bond equal 1.47.
Along with formic acid complexes, we have performed the calculations of electronic structure for monomer and dimer of acrylic
acid and its complexes with water and dimethylsulfoxide (fig. 5).
Calculations of structure and energy of acrylic acid associates with one and two water molecules by semiempirical 1 method
have indicated that the complex with two water molecules is energy more preferable (formation energy by 1 method is equal to -
89.297 kcal/mol) in comparison with acid complex containing one water molecule (formation energy makes up -83.815 kcal/mol). Thus,
the formation energy of the complex with two water molecules is higher, by 5.5 kcal/mol, than that of the complex with one water
molecule.
Calculations of acrylic acid complex with two water molecules, carried out with geometry optimization in basis 3-21G, gave rise
to the cyclic structure shown in fig. 5. In this structure one water molecule forms hydrogen bond (bond length - 2.08 ) with carbonyl
group oxygen atom of the acid (O-H-bonds of this water molecule lie in the plane of acrylic acid), and oxygen atom of other water
molecule forms a hydrogen bond (bond length - 2.03 ) with H-O-group hydrogen atom of the acid (OH-bonds of this water molecule
are symmetric about generic plane). The hydrogen bond length between oxygen atom of the first water molecule and hydrogen atom of
the second one is 2.15 . Such association direction is coincident with that we observed above for formic acid complex with two water
molecules (fig. 4, IV). At the same time, we have also calculated the structure with two water molecules oriented to form hydrogen bond
with oxygen atom of acid group and hydrogen at
atom of =-bond (fig. 5, IX). According to the calculation results obtained by
semiempirical 1 method, such complexation direction is also energy equiprobable (formation energy is 205.762 kcal/mol, while
formation enthalpy for structure VIII (fig. 5) calculated by 1method is equal to -207.779 kcal/mol). Besides, for acrylic acid complex
with one water molecule, when water molecule is coordinated on the side of carbonyl group and the acid hydrogen atom, the
complexation is also energy-preferable.
Calculations of acrylonitrile complex with DMSO have given the results similar to those we observed for the complex of DMSO
with formic acid, namely, there are two types of complexes: the chain form as in the complex of formic acid with DMSO (fig. 4, VI), and
the cyclic structure (fig. 5, X). As for chain structure, hydrogen bond appears between the acid hydrogen atom and oxygen atom of S=O
group. In structure X (fig. 5) the complexation follows the direction in which the acid O-H-group forms hydrogen bond with oxygen
atom of dimethylsulfoxide S=O-group, and oxygen of carbonyl group C=O is symmetric about hydrogen atoms of both methyl groups of
dimethylsulfoxide. Acid molecule and S=O-bond lies in one plane dividing the angle between C-S-bonds in half.
Just as in case of formic acid associate with DMSO, the constant of magnetic shielding for those DMSO methyl group protons
that are closest to oxygen atom of acrylic acid carbonyl groups is less than the shielding value of two other methyl group protons farther
removed from oxygen atom of the acid S=O-group.
It has long been known that C-H groups forms weak hydrogen bonds. It is by this type of bonding that the formation and physical
properties of dimeric complexes chloroform - acetone Cl
3
C-H...O=C(CH
3
)
2
were explained (structure 1:1) [ 85 ]. The problem of
-.... interactions was considered by Steiner in detail [86]. Our calculations have indicated the presence of hydrogen bond -... in
the complexes of formic and acrylic acids with dimethylsulfoxide (fig. 4, V and fig. 5, X). This type of interaction manifests itself in
considerably higher energy of the complex formation. For cyclic formic acid complex (fig. 4, V), in which the process of binding
proceeds in two directions - by = and --bonds, formation energy is equal to -16.32 kcal/mol. This value is more than two times
greater in comparison with the chain structure of the complex (fig. 4, VI) wherein hydrogen bond exists only between the acid O-H-
groups and dimethylsulfoxide S=O-groups and formation energy makes up -7.92 kcal/mol.
The values of -....-bond length are adequate to the data given in [86] and equal to 2.1-2.3 . Of a sort of criterion
confirming -.... bonding is the calculation data for the constants of proton magnetic shielding. As for equilibrium geometrical
structure of the complex, the values of magnetic shielding constants of dimethylsulfoxide methyl group protons, most closely situated at
the oxygen atom of carbonyl bond of the acid, which are assumed to participate in -.... interaction, are less (28 ppm) than shielding
constants of two other hydrogen atoms of methyl groups (31 ppm). As it has been shown above, there exists quite typical reduction of the
constant of proton magnetic shielding for the protons of hydrogen-bonded systems.
Tables 3,4 and 5 demonstrate the dependence of Mallyken's charges (q), energy (a.u.) and magnetic shielding constants (, ppm)
for the complexes of formic and acrylic acids with dimethylsulfoxide on the selection of basis and calculation method.
Table 3. Dependence of Mallyken's charges (q), energy (a.u.) and magnetic shielding constants (, ppm) on the calculation
techniques for cyclic
1
complex of formic acid with dimethylsulfoxide. Atomic numbering is shown in fig. 4, V.
Basis C
1
S
2
H
3
H
4
H
5
C
6
O
7
H
8
H
9
H
10
O
11
C
12
H
13
O
14
H
15
q -0.708 0.882 0.228 0.290 0.199 -0.708 -0.853 0.290 0.228 0.199 -0.734 0.557 0.575 -.616 0.169 RHF/6-31G*
1
168.3 342.92 30.58 28.56 31.38 168.31 228.93 28.56 30.58 31.38 145.45 34.86 17.22 -13.93 24.81
q -0.647 0.737 0.211 0.259 0.185 -0.647 -0.690 0.259 0.211 0.185 -0.554 0.366 0.468 -0.471 0.127 B3LYP/6-31G*
2
152.6 200.81 30.32 28.10 30.84 152.67 187.44 28.10 30.32 30.84 105.53 36.13 17.92 -31.85 24.51
1
Energy -740.284005021 a.u.
2
Energy -742.95604158 a.u.
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 21
Table 4. Calculation data of energy, Mallyken's charges (q) and magnetic shielding (, ppm) for acrylic acid,
dimer (fig. (5, VII)) and the complex with dimethylsulfoxide (fig. (5, X)).
DMSO atoms
Basis
H
*
O-H C=O C
O=S CH
3
H
5,8
H
9,10
H
6,7
Energy
q 0.471 -0.739 0.779 -0.259 -0.571 -265.6513021 RHF/6-31G*//
RHF /3-21G 27.28 189.93 40.42 80.98 -49.23
q 0.412 -0.590 0.564 -0.118 -0.463 -267.157504
Acrylic
Acid B3LYP/6-31G*
27.31 157.50 39.06 71.77 -65.88
q 0.565 -0.763 0.822 -0.257 -0.697 -531.3264380
(VII)
RHF/6-31G*
19.79 164.30 30.22 80.74 49.95
q 0.577 -0.787 0.802 -0.259 -0.346 -0.853 -0.708 0.226 0.197 0.293 -817.1881825 RHF/6-31G*
// RHF/3-21G 17.38 170.93 31.76 77.89 69.05 227.28 168.37 30.61 30.72 28.45
q 0.469 -0.614 0.572 -0.117 -0.308 -0.690 -0.6476 0.209 0.183 0.262 -820.3587865
(X)
B3LYP/6-31G*
// RHF /3-21G 18.092 135.092 31.87 68.37 68.41 185.16 152.71 30.35 30.87 28.02
Note to the table: Basing on the data [51,52] for
13
CS in acrylic acid (in mass, without solvent) in scale from TMS (- 170.4
ppm,
alpha
127.2 ppm,
beta
131.9 ppm), for absolute values of shielding constants we obtain 24.65, 67.85, 63.15 ppm,
accordingly.
Table 5. Mallyken's charges (q) and nuclear magnetic shielding constants (, ppm) in the complex
of acrylic acid with two water molecules, calculated at RHF/6-31G // 3-21G level.
H
7
6
-
5
-
8
9
1
2
3
13
14
15
q 0.575 -0.789 0.822 -0.259 -0.344 0.535 -0.967 0.450 -0.983 0.542 0.454
17.84 174.28 33.32 78.27 68.39 25.23 31.10 31.01 30.67 23.13 30.94
To sum up, the proton CS in complex systems can be reliably calculated by Ditchfield's method [32] with all bases used in this
work, which makes it possible to predict the formation of associates and polymeric structures. Also, the calculated
13
chemical shifts
will convey the tendency for relative changes in going from one solution to another.
7. Influence of electric fields on chemical shifts.
All recent methods were approved for small molecules and their application to big molecular systems such as biological
molecules is as yet impossible. To interpret chemical shifts in these molecular systems such approximations as separate calculation of the
effects of magnetic anisotropy and electric fields generated by various polar molecular groups. Magnetic anisotropy effects are basically
allowed for with well-known Poppy-Connell formula [5] (see fig. 6).
Magnetic anisotropy and electric field effects:
(Mc. Connell, 1957).
E
= -AE
z
BE
2
(Buckingham, 1960).
Fig. 6 shows magnetic anisotropy effect of carbonyl and phenyl groups on chemical shifts of the nuclei located in their vicinity.
Sign (+) designates a shift into strong magnetic fields, and sign (-) - into weaker fields.
Taking into account the contributions of electric fields (intra- and intermolecular) is an actual problem both for estimating CS of
such big molecules as biological systems (as well as middle-sized molecules), and for studying medium effects.
It is known that electric fields may have a marked effect on nuclear magnetic shielding constant , that reveals itself in nuclear
resonant frequency shifting. The foundation for such interpretation is the dependence obtained by Buckingham [87], who considered the
influence of polar group in molecule on changes in C-H-bond proton. Based on qualitative reasoning, he has established that electric
field E, caused by the effect of polar groups both located in molecule, or belonging to neighbour solvent molecules, gives rise to proton
chemical shifts
in C-H-bond, equal to
E
z
kE E =
10
18 2
(2)
Where
z
- projection of vector E on bond direction, and value has the order -2.
Fig. 6. Magnetic anisotropy effect of carbonyl and phenyl groups on chemical shifts of nuclei located in their vicinity.
Sign (+) designates a shift into strong magnetic fields, and sign (-) - into weaker fields.
Full Paper _______________________________________________________________________________________ R.M. Aminova
22 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
Thus, the electric field directed from to H atom shifts the bond electrons from shielded H to atom, resulting in a shift of
proton signal into smaller values of magnetic field.
Formula (1) found wide application in interpreting NMR spectra, the square-law contributions in most cases being small. The
formalism taking into account electric fields and magnetic anisotropy effects on allows understanding the chemical shifts in such
complex molecular systems as protein molecules [88].
We have developed a variety of approaches allowing to estimate the influence of both magnetic anisotropy contributions and
electric field effects of polar groups on chemical shift.
Previously [89, 90] we obtained formulas for a linear variation of shielding constant in electric field on the basis of variational
method.
We have taken into account that the perturbation of molecular system is effected by three factors: external magnetic field H,
nuclear magnetic moment and electric field produced by polar group of molecule (or reaction field caused by medium impact). Thus
we have allowed for magnetic susceptibility and electric polarizability of the bond.
Wave function perturbed by external magnetic field , electric field and nuclear magnetic moment look like:
= + + +
0
1 1 1 ( )( )( ) f H g E
(3)
Here
0
is unperturbed ground-state wave function, which is the solution of Schrodinger equation in the absence of nuclear
magnetic moment and external fields, and
f g
,
, - unknown trial functions of electron radius-vector
k
r
r
, complex in the
general case.
In the case of axial symmetry
E
z
E
is distinct from zero, when electric field has nonzero component along the bond axis
(axis z), hence
(4)
Required correction functions have the appearance:
z
z a r z d b = + ( )
, ) (
1 1
z y b y a i f
x
+ = , z y ia g
x
=
2
(5).
Here d is selected so that the orthogonality condition for
z
0
and
0
could be fulfilled, which is needed for providing the
invariancy of polarizability expression in relation to the origin transfer. Taking the tensor components of polarizability a
zz
,
magnetic susceptibility
a
a
a
b
zz zz
= = 0
;
xx
a
2
0 =
;
xx xx
a b
1 1
0 = =
By substituting correction function (5) in formula (2), we obtain the following expression for linear
r
E field increment to tensor :
E
k k k
k
k
k
k
e
mc
r r r
r
M f =
2
2
0
2
3
0 0 0
2
.
.
4
2
0
3
0
2
0 0
M
r
g
h
m
f g
k
k
k
k
k
k
.
(6)
where
) (
2
k k k
r
mc
ieh
M =
r
(7).
The obtained formulas make possible the direct noniterative calculation of the characteristics of electric field effect on nuclear
magnetic shielding in axial symmetry molecular systems.
This results in the following expressions for diamagnetic
z
E
d
and paramagnetic
z
E
p
contributions into
E
:
d
E
H H H
z
e
mc
a
z r zZ
r
b
zr r zZ
r
d
r r zZ
r
=
+
+
+
+
|
\
|
.
|
|
(
(
4
3
2
2
0
2
3
0 0
2
3
0 0
2
3
0
( ) ( ) (
|
p
E
z
eh
mc
aa z ab z a b z r rz Z = + +
+
4
3
1 0
2
0 1 0
3
0 1 0
2
0 0 0
( ) (
+ + d rz dZ r ab z Z zy
0 0 0 0 1 0
2
0 0
2
0
) ( )
+ + + bb rz d rz rzy d ry
1 0
3
0 0
2
0 0
2
0 0
2
0
(
|
+
0
2
0 0 0
rz Z dZ rz )
+
+
2
2 0
2
3
0 0
2
3
0
eh
mc
ab
z z Z
r
zy
r
H H
[ (
( )
)
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 23
+
+
+
+ + bb
rz z Z
r
d
rz z Z
r
rzy
r
d
ry
r
H H H H
2 0
2
3
0 0
3
0 0
2
3
0 0
2
3
0
(
( ) ( )
)]
+ + +
2
1 2 0
2
0 1 2 0
2
0 0 0
h
m
aa b z a b b rz d rz [ ( )
+ + + ab b zy bb b rzy d ry
1 2 0
2
0 1 2 0
2
0 0
2
0
(
+ +
0
3
0 0
2
0 1 2 0
3
0
rz d rz ab b z ) )]}
|
p
E
z
eh
mc
aa z ab z a b z r rz Z = + +
+
4
3
1 0
2
0 1 0
3
0 1 0
2
0 0 0
( ) (
+ + d rz dZ r ab z Z zy
0 0 0 0 1 0
2
0 0
2
0
) ( )
+ + + bb rz d rz rzy d ry
1 0
3
0 0
2
0 0
2
0 0
2
0
(
|
+
0
2
0 0 0
rz Z dZ rz )
+
+
2
2 0
2
3
0 0
2
3
0
eh
mc
ab
z z Z
r
zy
r
H H
[ (
( )
)
+
+
+
+ + bb
rz z Z
r
d
rz z Z
r
rzy
r
d
ry
r
H H H H
2 0
2
3
0 0
3
0 0
2
3
0 0
2
3
0
(
( ) ( )
)]
+ + +
2
1 2 0
2
0 1 2 0
2
0 0 0
h
m
aa b z a b b rz d rz [ ( )
+ + + ab b zy bb b rzy d ry
1 2 0
2
0 1 2 0
2
0 0
2
0
(
+ +
0
3
0 0
2
0 1 2 0
3
0
rz d rz ab b z ) )]}
a
em
h
z b r
z
r
d
z
r
= +
(
(
2
2
0
2
0 0 0 0
2
0 0 0
b
e
ch
z y
z zZ
r
y
r
H H
1
0
2
0 0
2
0
0
2
3
0 0
2
3
0
1
=
+
+
)
d
rz
r
=
0 0
0 0
{ }
{ }
b
em
h
z r
z
r
d
z
r
rz d rz
r
z
r
d
z
r
r z d
=
+ +
+
2
3
2
0
2
0 0 0 0
2
0 0 0 0
2
0 0 0
0 0 0
2
0 0 0 0
2
0 0
2
0 0 0
a
e
hc
z zZ
r
H
1 0
3
0
=
+
b
e
hc
z y
z y
2
0
2 2
0
0
2 2
0
2
=
+
a
2
= 0
f i a y b yz
x
= + ( )
1 1
g i a y b yz
x
= + ( )
2 2
z
za r z d b = + ( )
Full Paper _______________________________________________________________________________________ R.M. Aminova
24 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
Based on these expressions and with our assistance, M.E. Filatov worked out the algorithm and the program in FORTRAN for
personal computer that made it possible to calculate shielding changes due to polarizing of the bond to which shielded nucleus [94, 96]
belongs.
Using variational method in combination with non-empirical wave functions unperturbed by magnetic field [89, 90], the
computational algorithm has been built up for linear, in electric field, correction to of X-Y bond random nucleus, which is produced by
polar group of molecule.
Some integrals in formulas (6-7) were solved in analytical form. The most complex integrals were solved by numerical method.
This has made possible the usage of Sletter's wave functions for describing shielded bond. Besides, for the description of intramolecular
bonds Boys's method for localizing canonical molecular orbitals has been used.
To check the efficiency of the developed computational algorithm
E
z
the test calculations on
13
C-type nuclei for individual
bonds have been performed with the use of standard Richardson's function. The calculation results are shown in table 6. and 7.
Table 6. Values of correction function parameters
for
1
H shielding in C-H bonds.
d a b A
1
b
1
b
2
SP
3
0.114 -1.467 -0.328 0.259 -0.029 0.148
SP
2
0.022 -1.276 -0.272 0.265 -0.042 0.097
SP 0.025 -1.265 -0.136 0.267 -0.040 0.030
(SP
3
) [5] 0.086 -1.178 -0.270 0.244 -0.037 0.086
Table 7. Corrections to
13
magnetic shielding in C-H and C--bonds and
to
1
in H-H-bond, caused by electric field effect.
E
d
z
|
\
|
.
|
E
p
z
|
\
|
.
|
E
z
|
\
|
.
|
E
z
|
\
|
.
|
[5, 6]
H-H 3.226 0.635 3.861 3.1
C-H 25.830 6.845 32.675 30
C-C 44.071 10.247 54.318 51
The comparison of the obtained data with the experimental ones has confirmed the validity of the developed algorithm for
calculating both
1
H, and
13
C-type nuclei. We have also calculated
z
E
for a number of molecules for which the reliable experimental
data are available. In our calculations we used wave functions of the molecules designed in basis STO-3G and localized by Boys'
method. Table 8 shows the calculation results for diamagnetic, paramagnetic and total contributions into linear, in electric field,
shielding constant variation in C-H, C-C, C-N, C-O and C-F bonds, given in comparison with the available literary data.
Table 8 shows the coefficient of shielding constant linear variation in electric field for various bonds under the effect of polar
molecular groups. These results were obtained for specific molecules by calculating their electronic structure and subsequent localization
of their canonical orbitals. For further investigation of CS tendencies it is essential to estimate the components of electric field intensity
of polar groups as to the direction of the bond to which the shielded proton belongs.
We proposed that the contributions of magnetic anysotropy groups into shielding change should be estimated by calculating the
effect of localized fragments on CS change. It is convenient to use variational methods allowing wave functions of the ground molecular
state to be applied.
The diagrams of magnetic isoshielding lines, calculated in the vicinity of various bonds and lone-electron pairs (LEP), are
convenient for routine practical applications. Such diagrams are given below for C-H, C-N and for LEP of oxygen and nitrogen atoms
(fig. 7, 8).
The relative variation of proton chemical shifts caused by the changes in heterocyclic part of dioxane derivatives are given in fig.
9, 10. Wave functions of unperturbed state in heterocyclic part of molecule were calculated by non-empirical methods, after which the
contribution into chemical shifts of axial and equatorial protons was calculated.
Table 8. Diamagnetic,
E
d
z
|
\
|
.
|
paramagnetic
E
p
z
|
\
|
.
|
and total
E
z
|
\
|
.
|
contributions into linear,
in electric field, shielding constant variation for the bonds in studied molecules.
Molecule Bond
E
d
z
|
\
|
.
|
E
p
z
|
\
|
.
|
E
z
|
\
|
.
|
E
z
|
\
|
.
|
[6,7]
C-H* 3.22 0.63 3.85
CH
4
C-H* 5.84 0.93 6.77
C*-H 20.24 5.42 25.66 30.0
C-H* 6.11 0.38 6.49
CH
3
F C*-H 24.29 7.25 31.81
C*-F 51.37 16.08 67.45
C-F* 82.17 19.33 101.5 Cl=120.0 Br=82.0
CH
3
OH C-H* 5.96 0.75 6.71
C-O* 61.52 15.04 76.56
C*-H 22.92 6.18 29.80
CH
3
-CH
3
C-H* 4.77 0.51 5.28
C*-H 20.95 6.23 27.18
C-C* 44.97 10.22 55.19 51.0
C-H* 5.36 1.19 6.55
CH
3
COH C*-H 22.71 4.45 27.16
C*-O 58.40 16.01 74.41
C-H* 5.02 0.98 6.00
CH
3
NH
2
C*-H 22.33 1.59 23.92
C*-N 36.21 7.5 43.71 45.0
C-H* 5.68 1.46 7.14
CH
3
NO
2
C*-H 23.94 7.20 31.14
C*-N 40.69 8.3 48.99
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 25
Fig. 7. Diagram of isoshielding lines for lone-electron pair in aliphatic amines.
Fig. 8. Diagram of isoshielding lines for -N-bond in aliphatic amines.
Fig. 9. Diagram of isoshielding lines for C-H-bond in alkanes.
Fig.10. Diagram of isoshielding lines for lone-electron pair in formaldehyde.
Fig. 11. Relative variation of proton chemical shifts at the 4,6-position
of 1,3-dioxane derivatives depending on the type of heterocyclic fragment of molecule.
Then, the expressions for diamagnetic and paramagnetic contributions into shielding take the following form:
Full Paper _______________________________________________________________________________________ R.M. Aminova
26 ________________ http://chem.kstu.ru _________________ . . 2002. T.2. 6. 11.
Table 9, 10 exhibit the calculation data on magnetizability and magnetic shielding for a number of small molecules, obtained by
this method.
Table 9. Magnetizability calculations.
Molecule
d
d
Karplus
[13]
p
Karplus
[13]
< > < >exp
C*H
4
CY*
4
HF*
Li*H
LiH*
Li*F
LiF*
F
2
H
2
O
H*
2
O
PH*
3
H
2
S
-30.319
-65.058
-9.803
-22.501
-25.952
-75.435
-30.382
-64.99
-14.103
-37.474
-9.588
-20.529
-24.687
-76.139
-31,432
-64.800
11.730
46.668
0.774
11.284
14.654
59.588
14.756
45.02
3.792
27.885
0.937
11.38
14.67
60.98
15.29
46.15
-18.588
-18.390
-9.725
-11.217
-11.299
-27.710
-15.650
-19.96
-10.310
-9.5588
26.74
23.06
-18.7
-8.6
-10.10
-15.885
-18.6
-10.4
26.23
25.52
Table 10. Calculation data on proton magnetic shielding in basis STO-6G.
Molecule
H
d
H
p
H
Aminova
H
Holler, Lischka
H
Ditchfield
Relative
CH
4
Exp
CH
4
C
2
H
2
C
2
H
4
C
2
H
6
NH
3
H
2
O
HCN
88.47
99.28
110.35
155.38
97.11
102.51
99.46
56.29
67.75
83.47
125.33
64.91
73.10
69.72
32.18
31.53
26.88
30.05
32.20
29.41
29.74
31.39
29.85
25.42
30.63
31.41
30.13
---
32.73
31.48
27.12
32.38
33.59
32.79
29.78
0
-0.65
-5.29
-2.13
0.02
-2.57
-2.44
0
-1.35
-5.18
-0.75
0.05
-0.60
-2.83
Within the framework of variation-perturbation theory and using Karplus's formalism [13, 14], V.B. Mushkin has elaborated,
with our assistance, the program for shielding calculations in the systems with axial symmetry. This method permits the use of wave
functions of the molecular system ground-state, unperturbed by magnetic field, for shielding constant calculations. In our opinion, the
possibility of chemical shift calculations with ground-state wave functions is of interest in studying relative influences of molecular
fragments on chemical shift variation in complex molecular systems, such as biological objects. According to [20, 21], in such systems
the methods based on the combination of magnetic anisotropic and electric field effects of different polar group are effective. It should be
noted here that more exact methods for such molecules as proteins etc. is not a practical task at present because of numerous degrees of
freedom, and molecular geometry cannot be calculated with desired precision. For example, constants of magnetic shielding are very
sensitive to bond lengths. The aim of this work was to analyze the possibilities of variation-perturbation method in uncoupled variant
with wave functions of different precision for chemical shift calculations.
The perturbed wave function has the appearance:
(
+ + =
i
g
mc
e
i
f B
mc
e
o
i i N
2
1
1
(8).
Here
o
i i
f B mc e ) )( 2 / (
and
o
i i
g mc e
N
) )( / (
represent the polarization of the molecular orbital
o
i
by
magnetic field B and magnetic moment
N
, respectively, f
i
and g
i
are the variation functions depending on electron
coordinates.
For the origin of magnetic vector potential chosen on magnetic nucleus we have the final equation for paramagnetic contribution
into shielding tensor
}
1
*
3
* * 0 0
4
*
2
3
*
2
*
2
1
{
2 2
2
=
(
(
(
+ + |
.
|
\
|
|
.
|
\
|
+
|
.
|
\
|
+
=
= |
.
|
\
|
n
j
i m j j
i
g i i
N
r
m
j j
i
f i i
i
g j j
i
f i
i j
i
i
g m i i
N
r
i
f m
i i
i
g
i
f i
m
n
i
c m
e
N
p
h
(9).
Here <i, and ,i> signify o
i
and o
i
, correspondingly.
Variational functions f
i
and g
i
are written as:
) (
) (
N
N
N
N
N
N N N N N
z
i
D
i
C
r
z
i
B
r
i
A
iy
ix
g
r z
i
d r
i
c z
i
b
i
a iy
ix
f
+ + + =
+ + + =
(10).
Parameters of trial functions f
ix
were defined by solving Eulers equations, obtained by finding the stationary values of functional
xx
N
p
) (
.
0
) ( ) ( ) ( ) (
=
i
D
xx
N
p
i
C
xx
N
p
i
B
xx
N
p
i
A
xx
N
p
(11)
QUANTUM-CHEMICAL METHODS IN THE CALCULATIONS OF NUCLEAR MAGNETIC SHIELDING CONSTANTS _____________________________ 11-30
Chemistry and Computational Simulation. Butlerov Communications. 2002. Vol.2. No.6. _____________ E-mail: info@kstu.ru ____________________ 27
Table 11. Comparison of calculated constants of nuclear magnetic shielding constants with the results obtained by other methods and by experiment.
This work
(6-311G)
GIAO CHF
IGLO LORG
MCSCF
CAS (basis III)
h)
SOS-DFT (basis III)
i
(exp)
H
2
26.67
27.03
p)
27.16
q)
26.56 [9]
H
*
F 32.43
30.18
k)
30.70
p)
31.61
q)
29.25
f)
27.85
f)
28.05
e)
29.99 29.5
31.1 [9]
29.2+0.5
a)
HF
*
446.20
412.4
k)
419.42
p)
415.40
q)
415.0
f)
413.5
e)
420.5 410.0
413.7
a)
410+6
a)
H
*
Li 27.12
26.34
p)
26.36
q)
26.62
f)
26.59
e)
HLi
*
96.90
93.04
p)
92.92
q)
89.53
f)
89.67
e)
Li
*
F 85.60
92.16
p)
89.47
q)
87.5 [9]
LiF
*
382.72
391.85
p)
347.09
q)
374.3 [9]
CH
*
4
31.26
31.83
m)
30.61
k )
31.25
e)
31.2 30.47+0.2
n )
C
*
H
4
195.31 192.9
m )
195.8
f)
193.9
e)
196.0
l )
198.7
199.8
191.9 195.1
d)
CH
*
3
F
27.71 26.4+0.2
n )
CH
3
F
*
465.13 484.5
k)
450.0
g)
406.0
g)
448.0 471.0
b)
C
*
H
3
F 137.37 124.0
g)
132.0
g)
107.9 132.0
d)
H
2
O
*
358.41
326.7
o)
328.1
f)
325.6 334.0
j )
H
*
2
O 28.37 32.04
k)
30.13
f)
30.01
o)
30.42 31.2 30.052+0.015
c)
Comments to table 11
a) Reference [23].
b) Reference [24].
c) Reference [22].
d) Reference [26].
e) Reference [19].
f) Reference [15]. The results obtained in the framework of conventional coupled Hartree-Fock (CHF) method [29] with (11s7p3d/6s3p) basis set. The results
are given for origin at (H) for
1
H magnetic shielding, at (C) for
13
C magnetic shielding.
g) Reference [27].
h) Reference [28].
i) Reference [7].
j) Reference [29].
k) Reference [25]. Chemical shifts have been calculated employing (6311/311/1)=[4s,3p,d] heavy atom, (3,1)=[2s] hydrogen basis in the GIAO SCF
perturbation theory approach [17].
l) Reference [18].
m) Reference [30]. Calculations of chemical shifts have been carried out in the GIAO SCF approach [17] using heavy atom triple split valence basis
(6311/311/1)=[4s,3p,d]. The hydrogen basis was the (31)=[2s] basis.
n) Reference [31, 3].
o) Reference [32].
p) Calculations were carried out in this work in 6-311++G basis set using GIAO [17] method and Gaussian 94 program [34].
q) Calculations were carried out in this work in B3LYP//6-311++G basis set using GIAO [17] method.
0
) ( ) ( ) ( ) (
=
i
d
xx
N
p
i
c
xx
N
p
i
b
xx
N
p
i
a
xx
N
p
(12).
In calculations of nuclear magnetic shielding constants with the use of ground-state Hartree-Fock functions, obtained with 6-31G
and 6-311G basis sets, we employed numerical methods.
The calculation results for the series of diatomic and small molecules are given in tab. 11. The comparison of our data with
experimental data and the results obtained by other authors shows their good agreement.
Thus, the application of ground-state wave functions makes this method suitable for the analysis of chemical shifts of big
molecules. The considered method can serve as a good alternative to empirical methods. It can be used for estimating the contributions of
various molecular fragments of big biological molecules to chemical shifts.
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