Calculation of nuclear magnetic shieldings. XI. Vibrational motion effects
H. Fukui, T. Baba, J. Narumi, H. Inomata, K. Miura, and H. Matsuda
Kitami Institute of Technology, 165 Koencho, Kitami 090, Japan
~Received 27 March 1996; accepted 10 June 1996!
Nuclear magnetic shieldings in first- and second-row hydrides were calculated with electron
correlation taken into account through third order. The calculation was performed using London’s
gauge-invariant atomic orbitals ~GIAOs! and finite-field Mo” ller–Plesset perturbation theory
~FF-MPPT!. Furthermore, the vibrational motion corrections to the magnetic shieldings were
evaluated. It was shown that the calculated isotropic shielding constants at the experimental
geometries are higher than the experimental values, but that vibrational corrections are generally
negative and improve the calculated shielding constants. © 1996 American Institute of Physics.
@S0021-9606~96!00835-5#
I. INTRODUCTION
II. RESULTS AND DISCUSSION
In the last few years, several gauge-invariant or at least
approximately gauge-invariant calculations of nuclear magnetic shieldings at electron-correlated levels have been
presented.1–12 Until several years ago, it was believed that
the coupled Hartree–Fock ~CHF! approximation, which includes only first-order electron correlation correction in
terms of the Mo” ller–Plesset ~MP! perturbation theory, would
give sufficiently good results for calculations of nuclear
magnetic shielding constants in most cases, e.g., first-row
hydrides.13 Recently we presented the second- and thirdorder MP correlated calculations for the nuclear magnetic
shieldings in four first-row hydrides HF, H2O, NH3 , and
CH4 , which were performed with the London’s gaugeinvariant atomic orbitals ~GIAO!.8,14 We showed that
second- and third-order correlation corrections are not so
small as estimated before.15
Sugimoto and Nakatsuji16 recently suggested that the
SCF result using GIAOs may show coordinate origin dependence. However, their prediction is incorrect because the
CHF calculation with GIAOs has been proved to be independent of any displacement of coordinate origin.17 A recent
development in the calculation indicates that the combination
of an electron-correlated method and use of GIAOs can afford the most reliable results for nuclear magnetic
shieldings.1–8
Sometimes experiments and theoretical calculations are
complementary to each other. In fact it is hard work to get
experimentally absolute values of the magnetic shielding
constants in the condition of an isolated molecule. On the
contrary, usual theoretical computations yield the absolute
shielding constant of an isolated molecule. Therefore, it is a
work of importance to perform accurate calculations for absolute magnetic shielding constants of small and simple molecules which may serve as a standard substance in NMR
chemical shift measurements. In this work, we present correlated calculations for the nuclear magnetic shieldings in
first- and second-row hydrides and estimate the vibrational
corrections to the shielding constants.
4692
J. Chem. Phys. 105 (11), 15 September 1996
A. Experimental geometry values for the magnetic
shieldings
We performed CHF, finite-field self-consistent-field ~FFSCF!, finite-field second- and third-order Mo” ller–Plesset
~FF-MP2 and FF-MP3! correlation calculations with GIAOs
for nuclear magnetic shieldings in four first-row hydrides
HF, H2O, NH3 , CH4 and four second-row hydrides HCl,
H2S, PH3 , SiH4 . The theory for these calculations of magnetic shielding constants is shown in our previous paper.8
The magnitudes of the finite-field parameters, i.e., magnitudes of magnetic flux density components and nuclear dipole moment components, were chosen appropriately from
the condition that the FF-SCF value should reproduce the
CHF value corresponding to analytical derivatives of the energy. The CHF value was obtained with the use of the density matrix method proposed by Dodds et al.18 We used the
basis sets proposed by Schäfer et al.,19 which were augmented by the addition of polarization functions.
The results for the calculation of nuclear magnetic
shieldings in the first- and second-row hydrides at the experimental geometries20 are shown in Tables I and II, respectively, with other calculated results and experimental
values.13,21–24 Table I shows that our MP3 results agree well
with the most recent GIAO coupled-cluster singles and
doubles ~GIAO-CCSD! results given by Gauss et al.3 The
calculated isotropic shielding values sa v for F in HF at the
electron-correlated levels are substantially higher than experimental values. The difference may be referred to as rovibrational effects on the magnetic shieldings, which will be
discussed in the following subsections.
To our knowledge, the electron-correlated calculations
for the magnetic shieldings in the second-row hydrides have
not been presented except the GIAO-MP2 shielding calculations for the P atom in PH3 .25 Table II shows that the isotropic shielding values decrease with an increasing quality of
the basis sets. It is apparent that the basis set used in our
MP3 calculations is insufficient for the second-row hydrides,
but improvement of our MP3 program will be necessary before we can use bigger basis sets. Tables I and II show that
the calculated magnetic shielding constants of the non-
0021-9606/96/105(11)/4692/8/$10.00
© 1996 American Institute of Physics
Fukui et al.: Nuclear magnetic shieldings. XI
4693
TABLE I. Nuclear magnetic shieldings ~in ppm! in first-row hydrides at the experimental geometries.a
[6s4 p2d/3s1 p]
HF
F
H
H2Od
O
NH3
H
N
CH4
H
C
H
s'
si
sa v
Ds
s'
si
sa v
Ds
sxx
sy y
szz
sa v
sa v
s'
si
sa v
Ds
sa v
sa v
s'
si
sa v
Ds
[8s5 p2d/3s2 p]
CHF
SCF
MP2
MP3
CHF
SCF
MP2
MP3
378.5
482.0
413.0
103.5
21.61
44.09
29.10
22.49
304.5
361.1
311.8
325.8
31.32
273.0
235.0
260.3
238.0
32.14
194.9
28.69
38.36
31.92
9.67
378.5
482.2
413.1
103.7
21.61
44.10
29.10
22.49
304.6
360.8
311.4
325.6
31.32
272.8
235.2
260.3
237.6
32.14
195.3
28.74
38.34
31.94
9.60
394.9
481.8
423.4
87.5
22.34
44.24
29.64
21.90
321.6
370.3
336.9
342.9
31.38
288.4
245.2
274.0
243.2
32.01
201.3
28.49
38.36
31.78
9.87
385.0
482.0
417.3
97.0
22.61
44.34
29.85
21.73
312.4
365.7
325.3
334.5
31.63
282.5
240.3
268.4
242.2
32.18
199.2
28.64
38.29
31.86
9.65
379.5
482.0
413.7
102.5
20.81
44.10
28.57
23.29
306.1
364.4
313.6
328.0
30.88
274.6
235.9
261.7
238.7
31.78
195.3
28.37
38.31
31.69
9.94
379.5
482.1
413.7
102.6
20.81
44.09
28.57
23.28
306.4
363.2
312.0
327.2
30.87
274.3
235.9
261.5
238.4
31.78
195.2
28.37
38.31
31.69
9.94
395.0
481.7
423.9
86.7
21.70
44.28
29.23
22.58
322.8
372.7
336.9
344.1
30.99
289.3
245.4
274.7
243.9
31.66
200.9
28.12
38.32
31.52
10.21
385.4
481.8
417.6
96.4
21.93
44.37
29.41
22.44
313.4
367.8
325.1
335.4
31.23
283.1
240.4
268.9
242.7
31.82
[10s7 p3d/4s2 p]
CHF
Other resultsb
379.3
481.9
413.5
102.6
20.91
44.09
28.64
23.17
304.8
364.8
313.0
327.5
30.93
274.2
235.2
261.2
239.0
31.80
195.0
28.41
38.28
31.70
9.88
Expt.c
418.1
95.0
41066
93.8b
29.1
22.8
28.72
336.9
30.9
344.0617.2b
30.09
269.7
243.2e
31.6
198.7
264.5
240.3f
30.68 ~liq!
197.4
31.5
10.1
30.61
a
Taken from Ref. 20.
Taken from Ref. 3.
c
Taken from Ref. 13 if not noted otherwise.
d
The z axis is parallel to the C 2 axis. The x axis is perpendicular to the molecular plane.
e
The original value 221.6 ppm in Ref. 3 was doubted.
f
Taken from Ref. 21.
b
hydrogen atoms rather well reproduce the experimental values, but the agreement between calculations and experiments
is rather poor for the proton magnetic shielding constants.
We will estimate vibrational corrections to the proton shieldings later.
B. Rovibrational corrections for the shieldings in
diatomic molecules
In diatomic molecules we have one vibration which
couples with the rotational motion. We assumed the following interatomic potential function V for diatomic molecules:
V5 ~ 1/2! KR 2e j 2 ~ 11a 1 j ! ,
j 5 ~ r2r e ! /r e ,
~1!
where K is the force constant and r e is the equilibrium value
for the bond length r. The averages of jn can be expressed in
terms of the parameter a 1 and the usual molecular constants
ve and B e , in cm21.26 We have27
S D S D
S DS D
^ j & v J 523 v 1
^ j 2 & v J 52 v 1
1
2
S D
1
Be
Be
a1
14J ~ J11 !
2
ve
ve
Be
,
ve
2
,
~2!
~3!
where v and J are the vibration and rotation quantum numbers, respectively. At room temperature we may assume
v 50. The rovibrational correction to the magnetic shielding
tensor s is given by
^ s & vJ2 s e5
S D
]s
]j
j 50
^ j & vJ1
S DS D
1
2
] 2s
]j 2
j 50
^ j 2 & v J , ~4!
where se is the magnetic shielding tensor at the equilibrium
bond length, i.e., at j50. We calculated the rovibrational
corrections to the magnetic shieldings in the HF and HCl
molecules at 300 K. The results for HF and HCl are shown in
Table III. Table III indicates that the rovibrational correction
improves our results for HF and HCl shown in Tables I and
II.
C. Vibrational corrections for the shieldings in C 2 v
molecules
For polyatomic molecules we neglected the effects of
rotational motions on the magnetic shieldings. It has been
shown that the effect of centrifugal distortion due to molecular rotational motions is usually one order of magnitude
smaller than the effect due to anharmonic vibrations.28 We
will calculate here only the vibrational effects on the magnetic shieldings at 0 K in the polyatomic molecules.
We first set up for the C 2 v molecules the three normal
coordinates from the internal coordinates Dr 1 , Dr 2 , and Du,
i.e., the bond length and bond angle deformations, according
to the standard method by Wilson.29,30 We calculated the
normal coordinates iteratively by using experimental values
for the normal mode vibration frequencies.31
J. Chem. Phys., Vol. 105, No. 11, 15 September 1996
Fukui et al.: Nuclear magnetic shieldings. XI
4694
TABLE II. Nuclear magnetic shieldings ~in ppm! in second-row hydrides at the experimental geometries.a
[8s6 p1d/3s1 p]
CHF
HCl
Cl
H
H2Se
S
PH3
H
P
SiH4
H
Si
H
s'
si
sa v
Ds
s'
si
sa v
Ds
sxx
sy y
szz
sa v
sa v
s'
si
sa v
Ds
sa v
sa v
s'
si
sa v
Ds
858.5
1149.4
955.4
290.9
23.45
45.37
30.76
21.92
688.5
925.2
570.6
728.1
31.22
608.0
569.9
595.3
238.1
29.85
489.3
26.19
33.31
28.56
7.12
SCF
859.4
1149.1
956.0
289.7
23.40
45.37
30.72
21.97
689.4
925.3
570.7
728.4
31.22
608.0
569.9
595.3
238.1
29.85
489.8
26.19
33.31
28.56
7.12
MP2
881.1
1149.1
970.5
268.0
24.04
45.43
31.17
21.39
712.2
936.5
634.3
761.0
31.39
633.7
580.2
615.9
253.5
29.89
490.3
26.16
33.35
28.56
7.19
[8s6 p2d/3s2 p]
MP3
868.2
1149.1
961.8
280.9
24.45
45.57
31.49
21.12
702.8
930.5
617.5
750.3
31.57
627.7
576.4
610.6
251.2
29.97
489.4
26.23
33.33
28.60
7.10
CHF
853.4
1149.1
952.0
295.7
23.47
45.61
30.85
22.14
678.5
929.3
555.6
721.3
31.16
598.9
559.6
585.8
239.3
29.75
477.8
26.05
33.09
28.40
7.04
SCF
853.6
1149.1
952.0
295.5
23.39
45.61
30.80
22.22
678.4
929.6
555.7
721.2
31.16
598.9
559.7
585.8
239.3
29.75
477.9
26.05
33.09
28.40
7.04
MP2
883.8
1148.9
972.1
265.1
23.82
45.60
31.08
21.78
708.1
941.2
635.3
761.5
31.28
631.0
571.5
611.2
259.5
29.77
476.3
25.89
33.07
28.28
7.18
[10s7 p3d/4s2 p]
CHF
MP3
869.1
1148.9
962.3
279.8
24.21
45.74
31.39
21.53
696.9
934.4
613.8
748.4
31.44
624.0
567.1
605.1
256.8
29.81
Other results
Expt.
b
849.5
1149.1
949.4
299.6
23.34
45.55
30.75
22.21
675.0
931.0
551.1
719.0
31.08
596.4
557.2
583.4
239.2
29.71
477.4
26.07
32.97
28.37
6.91
859.75
1148.93b
956.14b
289.18b
32.25d
952b
298c
31.5d
688.15b
936.64b
570.89b
731.89b
32.69d
629.61f
565.07f
608.09f
264.54f
30.96d
482.47b
752b
30.54d
621.43f
556.93f
599.93f
264.50f
28.3d
475.3b
29.32d
27.61d
a
Taken from Ref. 20.
Taken from Ref. 22.
c
Taken from Ref. 23.
d
Taken from Ref. 24.
e
The z axis is parallel to the C 2 axis. The x axis is perpendicular to the molecular plane.
f
Taken from Ref. 25.
b
Furthermore, we defined the three internal symmetry coordinates:
S 1 5 ~ Dr 1 1Dr 2 ! /&,
S 2 5r e D u /&,
S 3 5 ~ Dr 1 2Dr 2 ! /&,
where r e is the equilibrium bond length. The internal symmetry coordinates S are expressed in terms of the normal
coordinates Q as S5LQ. Namely,
F G F GF G
a
S1
S2 5 g
S3
0
b
0
d
0
0
e
Q1
Q2 ,
Q3
~5!
where Q 1 , Q 2 , and Q 3 are A 1 stretching, A 1 bending, and B 2
stretching normal coordinates, respectively. Numerical values for the matrix elements of L are given in Table IV for
H217O and H233S. Off diagonal elements of the matrix L are
less than one tenth of the diagonal elements.
One of the important applications of the normal coordinates is the calculation of the mean-square values of the internal symmetry coordinates. By ignoring contributions from
anharmonic potentials the mean values of the quadratic products of the normal coordinates at 0 K are given by
^ Q i Q j & 5 ~ \/2v i ! d i j ,
~6!
where vi is the angular frequency of the ith normal mode
vibration. The mean values of the quadratic products of the
symmetry coordinates ^ S i S j & are readily evaluated with Eqs.
~5! and ~6!.
In order to obtain the mean values of the S 1 and S 2
coordinates with A 1 symmetry we express the potential energy V for the nuclear motions in the molecule in terms of
internal symmetry coordinates S by the relation
2V5
(i (j F i j S i S j 1 (i (j (k F i jk S i S j S k ,
~7!
where the F i j and F i jk represent quadratic and cubic force
constants. The equilibrium potential V e is set here to be 0.
The mean values of the symmetry coordinates ^ S i & are determined by the equilibrium conditions32
K L
]V
50.
]Si
~8!
The vibrational correction to the magnetic shielding tensor is
given by
^ s & 0 K2 s e 5 (
i
S D
]s
^S &
]Si e i
11/2
J. Chem. Phys., Vol. 105, No. 11, 15 September 1996
(i (j
S
D
] 2s
^S S &.
] S i] S j e i j
~9!
Fukui et al.: Nuclear magnetic shieldings. XI
TABLE III. Rovibrational corrections to the isotropic shielding constants
and shielding anisotropies in H19F and H35Cl at 300 K.
H19F
ve a ~cm21!
B e a ~cm21!
a 1a
~]sa v /]j!j50 ~ppm!b
F
H
F
H
F
H
F
H
~]2sa v /]j2!j50 ~ppm!b
~]Ds/]j!j50 ~ppm!b
~]2Ds/]j2!j50 ~ppm!b
^j&b
^j2&b
^sa v &300 K2sea v ~ppm!b
F
H
F
H
^Ds&300 K2Dse ~ppm!b
H35Cl
ve a ~cm21!
B e a ~cm21!
a 1a
~]sa v /]j!j50 ~ppm!c
~]Ds/]j!j50 ~ppm!c
~]2Ds/]j2!j50 ~ppm!c
^j&c
^j2&c
^sa v &300 K2sea v ~ppm!c
^Ds&300 K2Ds ~ppm!
2989.7
10.59
22.341
2877.8
241.62
22949.6
119.3
1314.7
246.61
4434.8
73.10
1.3431022
3.5431023
217.0
20.35
25.5
20.50
Cl
H
Cl
H
Cl
H
Cl
H
~]2sa v /]j2!j50 ~ppm!c
e
4183.5
20.939
22.211
2390.0
240.90
21205.2
122.9
580.2
249.10
1831.2
83.10
1.7831022
5.0631023
210.0
20.42
14.9
20.66
Cl
H
Cl
H
c
a
Taken from Ref. 26.
b
CHF calculations with the [8s5p2d/3s2 p] basis used.
c
CHF calculations with the [8s6p2d/3s2 p] basis used.
for sa v ~O! becomes 323.3 ppm. Our vibrationally corrected
value is a little lower than the experimental lower limit
shown by the error bar. We note that the A1 stretching vibration has the largest effect in the three vibrational motions.
Our value for the vibrational correction, 212.1 ppm, is
slightly smaller than the value reported by Fowler and
Raynes,33 213.1 ppm. The vibrational correction for the S
atom, 219.6 ppm, is about twice as large as that for the O
atom. The vibrationally corrected value for s a v (S) becomes
728.8 ppm, which is considerably lower than the experimental value 752 ppm.
D. Vibrational corrections for the shieldings in C 3 v
molecules
The NH3 and PH3 molecules have six vibrational motions. We used as the bases for the normal coordinates the
following
six
internal
symmetry
coordinates:
S 1 5(Dr 1 1Dr 2 1Dr 3 )/), S 2 5r e (D u 121D u 231D u 31)/),
S 3 5(Dr 2 2Dr 3 )/&,
S 4 5r e (D u 312D u 12)/&,
S 5 5(Dr 2 1Dr 3 22Dr 1 )/ A6, S 6 5r e (D u 121D u 3122D u 23)/
A6. The symmetry coordinates S 1 and S 2 have the A1 symmetry, but the coordinates S 3 to S 6 have the E symmetry.
Furthermore, the coordinates S 3 and S 4 are equivalent to the
coordinates S 5 and S 6 , and the S 3 and S 4 have no cross terms
between the S 5 and S 6 in any energy functions. The internal
symmetry coordinates S are expressed in terms of the normal
coordinates Q as S5LQ. The matrix L has the structure
represented by
F G
a
L5
0
b
0
~10!
,
b
where both a and b are the 232 matrices. We write here the
a and b as
a5
We calculated the vibrational corrections to the isotropic
shielding constants in H217O and H233S. We made calculation of the potential values at 125 points using the SCF
method and determined the force constants by the leastsquare method. The equilibrium conditions, Eq. ~8!, yielded
^S 1& and ^S 2&. The shielding derivatives were calculated as
the numerical derivatives of the shielding values obtained
with the CHF method.
The results for H2O and H2S are listed in Tables V.
Table V shows that the total vibrational correction to the
isotropic O shielding in H2O due to the three vibrational
motions is 212.1 ppm and our vibrationally corrected value
TABLE IV. Internal symmetry coordinates ~in atomic mass units!.
H217O
H233S
4695
a
b
g
d
e
1.0213
1.0143
0.0331
0.0199
20.0893
20.0498
1.0323
1.0144
1.0361
1.0156
F G F G
a
b
g
d
,
b5
e
z
h
q
.
~11!
Numerical values for the matrix elements of L are given in
Table VI for 15NH3 and 31PH3 .
We calculated the vibrational corrections to the isotropic
shielding constants and the shielding anisotropies in 15NH3
and 31PH3 . We made calculations of the potential values at
225 points using the SCF method and determined the force
constants by the least-squares method. The equilibrium conditions yielded ^S 1& and ^S 2&. The shielding derivatives were
calculated as the numerical derivatives of the shielding values obtained with the CHF method. The results are shown in
Tables VII and VIII. Table VII shows that the total vibrational correction to the isotropic N shielding in NH3 due to
the six vibrational motions is 27.0 ppm. Our value is a little
smaller than the value reported by Jameson and de Dias,34
28.8 ppm at 300 K. Our value for the vibrational correction
to the isotropic P shielding in PH3 is 27.3 ppm, which is
much smaller than the value by Jameson and de Dias,35
212.78 ppm at 300 K. Our calculation gave 28.51 ppm as
the ( ]s a v / ] S 1 ) e ^ S 1 & contribution, which is comparable to
J. Chem. Phys., Vol. 105, No. 11, 15 September 1996
Fukui et al.: Nuclear magnetic shieldings. XI
4696
TABLE V. Shielding derivatives and vibrational corrections to the isotropic shielding constants in H217Oa and
H233Sb at 0 K.
H217O
17
1
O
( ]s a v / ] S 1 ) e ~ppm/Å!
( ]s a v / ] S 2 ) e ~ppm/Å!
( ] 2 s a v / ] S 21 ) e ~ppm/Å2!
( ] 2 s a v / ] S 22 ) e ~ppm/Å2!
( ] 2 s a v / ] S 23 ) e ~ppm/Å2!
( ] 2 s a v / ] S 1 S 2 ) e ~ppm/Å2!
( ]s a v / ] S 1 ) e ^ S 1 & ~ppm!
( ]s a v / ] S 2 ) e ^ S 2 & ~ppm!
1/2( ] 2 s a v / ] S 21 ) e ^ S 21 & ~ppm!
1/2( ] 2 s a v / ] S 22 ) e ^ S 22 & ~ppm!
1/2( ] 2 s a v / ] S 23 ) e ^ S 23 & ~ppm!
( ] 2 s a v / ] S 1 S 2 ) e ^ S 1 S 2 & ~ppm!
Total correction ~ppm!
H233S
33
H
2372.9
28.795
2987.6
285.2
21041.8
4.026
28.78
0.00
22.38
1.61
22.51
20.00
212.1
1
S
229.94
25.503
70.00
25.801
26.69
20.5608
20.705
0.001
0.169
20.033
0.064
0.000
20.50
H
2597.6
268.07
21113.4
388.1
21414.7
228.54
214.09
20.03
23.70
2.85
24.68
0.00
219.6
220.26
23.057
29.09
23.665
9.477
1.601
20.477
20.001
0.097
20.027
0.031
20.000
20.38
a
[8s5 p2d/3s2p] basis used.
[8s6 p2d/3s2p] basis used.
b
their value, 29.16 ppm. However, our value for the
( ]s a v / ] S 2 ) e ^ S 2 & was 1.40 ppm, which disagrees in sign
with their value, 23.29 ppm. Our vibrationally corrected
shielding constants for N and P are 261.9 ppm and 597.8
ppm, respectively. Calculated values are improved by adding
the vibrational corrections. The vibration effects were also
evaluated for the isotropic proton shielding constants. The
protons have only the local symmetry lower than the molecular symmetry, C 3 v . However, the average magnetic shielding
for the three protons has the same symmetry as the molecule.
Table VIII shows that the vibrational motions have large
effects on the magnetic shielding anisotropies in NH3 and
PH3 . Our vibrationally corrected shielding anisotropies are
considerably smaller than the experimental values, especially
for the Ds~P! in PH3 . This disagreement may be due to large
contributions of high-order correlations in the shielding
anisotropies.
dinates with a use of the well-known method of the
group theory. The nine internal symmetry coordinates
are as follows: S 1 (A 1 )5(Dr 1 1Dr 2 1Dr 3 1Dr 4 )/2,
S 2 (E)5r e (2D u 122D u 132D u 142D u 232D u 2412D u 34)/ A12,
S 3 (E)5r e (D u 132D u 142D u 231D u 24)/2,
S 4 (T2 )5(Dr 1 2Dr 2 1Dr 3 2Dr 4 )/2,
S 5 (T2 )5r e (2D u 131D u 24)/&,
S 6 (T2 )5(Dr 1 2Dr 2 2Dr 3 1Dr 4 )/2,
S 7 (T2 )5r e (2D u 141D u 23)/&,
S 8 (T2 )5(Dr 1 1Dr 2 2Dr 3 2Dr 4 )/2,
S 9 (T2 ) 5 r e ( 2 D u 12 1 D u 34)/&. The three pairs of coordinates, i.e., (S 4 ,S 5 ), (S 6 ,S 7 ), and (S 8 ,S 9 ), are equivalent to
each other. The three pairs of coordinates have no cross
terms among them in any energy functions. This greatly reduces the number of independent force constants.
The internal symmetry coordinates S are expressed in
terms of the normal coordinates Q as S5LQ. The matrix L
has the structure represented by
E. Vibrational corrections for the shieldings in T d
molecules
A T d -type molecule like CH4 has ten internal coordinates, i.e., four bond length displacements and six bond
angle deformations. However, the number of degrees of freedom of vibrational motions is nine and the ten internal coordinates are evidently redundant. In fact, the sum of the six
bond angle deformations is zero. We have to drop a redundant coordinate without spoiling the symmetry of the coordinates. We can construct the nine internal symmetry coor-
L5
3
a
b
0
b
a
0
a
a
4
~12!
,
where a is the 232 matrix. We write here the a matrix as
TABLE VI. Internal symmetry coordinates ~in atomic mass units!.
15
NH3
PH3
31
a
b
g
d
e
z
h
q
1.0141
1.0140
0.0080
0.0193
20.0847
20.0864
1.1805
1.4130
1.0419
1.0169
20.0171
20.0120
0.1367
0.0527
1.5932
1.4483
J. Chem. Phys., Vol. 105, No. 11, 15 September 1996
Fukui et al.: Nuclear magnetic shieldings. XI
4697
TABLE VII. Shielding derivatives and vibrational corrections to the isotropic shielding constants in
and 31PH3 b at 0 K.
15
( ]s a v / ] S 1 ) e ~ppm/Å!
( ]s a v / ] S 2 ) e ~ppm/Å!
( ] 2 s a v / ] S 21 ) e ~ppm/Å2!
( ] 2 s a v / ] S 22 ) e ~ppm/Å2!
( ] 2 s a v / ] S 1 S 2 ) e ~ppm/Å2!
( ] 2 s a v / ] S 23 ) e ~ppm/Å2!
( ] 2 s a v / ] S 24 ) e ~ppm/Å2!
( ] 2 s a v / ] S 3 S 4 ) e ~ppm/Å2!
( ]s a v / ] S 1 ) e ^ S 1 & ~ppm!
( ]s a v / ] S 2 ) e ^ S 2 & ~ppm!
1/2( ] 2 s a v / ] S 21 ) e ^ S 21 & ~ppm!
1/2( ] 2 s a v / ] S 22 ) e ^ S 22 & ~ppm!
( ] 2 s a v / ] S 1 S 2 ) e ^ S 1 S 2 & ~ppm!
( ] 2 s a v / ] S 23 ) e ^ S 23 & ~ppm!
( ] 2 s a v / ] S 24 ) e ^ S 24 & ~ppm!
2( ] 2 s a v / ] S 3 S 4 ) e ^ S 3 S 4 & ~ppm!
Total correction ~ppm!
PH3
1
N
2218.6
213.51
2336.4
118.7
238.42
2540.7
66.60
27.436
26.59
0.12
20.87
1.47
0.01
22.90
1.76
20.01
27.0
NH3 a
31
NH3
15
15
31
H
1
P
223.12
23.216
37.21
20.2681
20.3409
12.15
21.959
20.05094
20.697
0.028
0.097
20.003
0.000
0.065
20.052
20.000
20.56
H
2274.0
2142.7
2215.9
164.4
278.20
2646.9
79.21
0.2297
28.51
1.40
20.80
2.80
0.01
24.66
2.50
0.00
27.3
212.91
21.924
13.68
20.9073
0.6595
5.460
21.813
0.8634
20.401
0.019
0.051
20.015
20.000
0.039
20.057
0.000
20.36
a
[8s5 p2d/3s2p] basis used.
[8s6 p2d/3s2p] basis used.
b
a5
F G
g
d
e
z
~13!
.
Numerical values for the matrix elements of L are given in
Table IX for 13CH4 and 29SiH4 .
We calculated the vibrational corrections to the isotropic
shielding constants in 13CH4 and 29SiH4 . We made calculation of the potential values at 135 points using the SCF
method and determined the force constants by the leastsquare method. The equilibrium condition yielded ^S 1&. The
shielding derivatives were calculated as the numerical deTABLE VIII. Shielding derivatives and vibrational corrections to the magnetic shielding anisotropies in 15NH3 a and 31PH3 b at 0 K.
15
NH3
N
15
( ] D s / ] S 1 ) e ~ppm/Å!
( ] D s / ] S 2 ) e ~ppm/Å!
( ] 2 D s / ] S 21 ) e ~ppm/Å2!
( ] 2 D s / ] S 22 ) e ~ppm/Å2!
( ] 2 D s / ] S 1 S 2 ) e ~ppm/Å2!
( ] 2 D s / ] S 23 ) e ~ppm/Å2!
( ] 2 D s / ] S 24 ) e ~ppm/Å2!
( ] 2 D s / ] S 3 S 4 ) e ~ppm/Å2!
( ] D s / ] S 1 ) e ^ S 1 & ~ppm!
( ] D s / ] S 2 ) e ^ S 2 & ~ppm!
1/2( ] 2 D s / ] S 21 ) e ^ S 21 & ~ppm!
1/2( ] 2 D s / ] S 22 ) e ^ S 22 & ~ppm!
( ] 2 D s / ] S 1 S 2 ) e ^ S 1 S 2 & ~ppm!
( ] 2 D s / ] S 23 ) e ^ S 23 & ~ppm!
( ] 2 D s / ] S 24 ) e ^ S 24 & ~ppm!
2( ] 2 D s / ] S 3 S 4 ) e ^ S 3 S 4 & ~ppm!
Total correction ~ppm!
a
[8s5 p2d/3s2 p] basis used.
b
[8s6p2d/3s2 p] basis used.
61.79
236.36
254.2
2140.2
31.46
179.6
147.9
121.4
1.86
0.31
0.66
21.74
20.01
0.96
3.90
0.10
6.1
31
PH3
P
31
132.0
42.24
10.39
2215.0
210.0
259.1
350.5
292.0
4.10
20.42
0.04
23.66
20.04
1.87
11.06
0.07
13.0
rivatives of the shielding values obtained with the CHF
method. The results are shown in Table X. Table X shows
that the total vibrational correction to the isotropic C shielding in CH4 due to the nine vibrational motions is 24.7 ppm.
Our vibrationally corrected value, 194.5 ppm, is a little lower
than the experimental value, 197.4 ppm. Table X shows that
the vibrational correction to the Si atom is small, i.e., 22.0
ppm. Our MP3 value for the isotropic Si shielding at the
experimental geometry is 489.4 ppm with the [8s6p1d/
3s1p] basis, which is too high compared with the experimental value 475.3 ppm. However, the use of the [8s6p2d/
3s2p] basis lowers the SCF and MP2 values by 12 to 14
ppm from the [8s6 p1d/3s1 p] values. So we may expect
that the MP3 value of larger bases would be around 477
ppm. The vibrational correction 22.0 ppm may be a reasonable value.
To our knowledge, the present article is the first report
on the vibrational effect calculation for the magnetic shieldings in T d -symmetry molecules. Jameson and Osten36 assumed in the calculation of the vibrational corrections to the
19
F shieldings in CHn F42n -type molecules that for the nearly
tetrahedral molecules the contribution of the bond angle deformations and the second-order derivative contribution of
the bond lengths can be neglected. On the other hand,
Enevoldsen and Oddershede37 asserted that for CH3F shieldings the second-order derivative contributions of the bond
TABLE IX. Internal symmetry coordinates ~in atomic mass units!.
13
CH4
SiH4
29
a
b
g
d
e
z
1.0000
1.0000
1.7321
1.7321
1.0396
1.0205
20.1476
20.0668
0.4099
0.1861
1.4974
1.4660
J. Chem. Phys., Vol. 105, No. 11, 15 September 1996
Fukui et al.: Nuclear magnetic shieldings. XI
4698
TABLE X. Shielding derivatives and vibrational corrections to the isotropic
shielding constants in 13CH4 a and 29SiH4 at 0 K.b
13
13
C
29
CH4
SiH4
1
H
29
Si
( ]s a v / ] S 1 ) e ~ppm/Å!
2104.0
217.34
241.50
( ] 2 s a v / ] S 21 ) e ~ppm/Å2!
234.46
19.90
279.32
( ] 2 s a v / ] S 22 ) e ~ppm/Å2!
17.52
22.236
5.535
( ] 2 s a v / ] S 24 ) e ~ppm/Å2!
2180.5
7.817 2257.9
( ] 2 s a v / ] S 25 ) e ~ppm/Å2!
49.17
0.2369
61.38
( ] 2 s a v / ] S 4 S 5 ) e ~ppm/Å2!
5.508
0.1553
3.914
( ]s a v / ] S 1 ) e ^ S 1 & ~ppm!
25.65
20.942
22.50
1/2( ] 2 s a v / ] S 21 ) e ^ S 21 & ~ppm!
20.10
0.057
20.31
( ] 2 s a v / ] S 22 ) e ^ S 22 & ~ppm!
0.58
20.074
0.29
3/2( ] 2 s a v / ] S 24 ) e ^ S 24 & ~ppm!
21.71
0.074
23.14
3/2( ] 2 s a v / ] S 25 ) e ^ S 25 & ~ppm!
2.20
0.011
3.69
3( ] 2 s a v / ] S 4 S 5 ) e ^ S 4 S 5 & ~ppm!
20.01
20.000
20.00
Total correction ~ppm!
24.7
20.87
22.0
a
1
H
28.784
5.151
21.042
3.969
20.381
0.4701
20.529
0.020
20.054
0.048
20.229
20.000
20.54
[6s4 p2d/3s2 p] basis used.
[8s6p2d/3s2 p] basis used.
b
lengths should not be ignored. We note that bond angle vibration contributions nearly cancel each other and note that
the second-order derivative contribution of S 1 is very small.
Our results show that the assumption by Jameson and Osten
may be a reasonable approximation.
F. Comparison between calculations and experiments
Tables I and II show that the MP3 values for the isotropic shieldings of the heavier atoms ~C to Cl! at the experimental geometries are higher than the experimental values.
However, the vibrational corrections are negative and improve the calculated shieldings of the heavier nuclei except
17
O and 33S. In H2O and H2S the MP3 shieldings are lower
FIG. 1. Calculated and experimental proton shieldings. ~s! vibrationally
corrected shieldings; ~d! MP3 shieldings at the experimental geometries.
than the experimental shieldings and the vibrational corrections further deteriorate agreement with the experimental
values. Gauss suggested that the experimental value 344.0
617.2 ppm for the isotropic O shielding in H2O may also be
too high.38 The experimental value for the isotropic S shielding in H2S may also be too high.
Tables I and II show that the MP3 values for the isotropic proton shieldings in the experimental geometry molecules are higher than the experimental shieldings except
HCl. The vibrational corrections to the isotropic proton
shieldings are negative and improve the calculated values.
The calculated MP3 proton shieldings at the experimental
geometries and the vibrationally corrected proton shieldings
are plotted against the experimental values in Fig. 1. Figure 1
shows that the vibrationally corrected proton shieldings are
yet higher than the experimental shieldings. The higher-order
correlation corrections and rotational motion corrections may
be necessary for obtaining better agreement with experimental proton shieldings.
ACKNOWLEDGMENTS
All the calculations were performed on a Convex C3440
at the Kitami Institute of Technology Computing Center. It is
a pleasure to express our appreciation to Professor F. Sasaki
for providing us with computer programs for molecular integrals.
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