36
2 Computational Methods
Table 2.18 Mean, standard deviation (St. dev.), root-mean-square error (RMSE) and standard
error of estimate (SEE) for the B2PLYP/VTZ and B3LYP/SNSD deviations from B2PLYP/VTZ
semiexperimental equilibrium structures (distances in pm and angles in degree)
CH
CC
CO
CN
∠(HCH)
∠(CCC)
n a
82
45
30
14
29
22
B2PLYP Mean 0.02
0.01
0.27
0.015
-.St. dev. 0.11
0.28
0.15
0.018
-.RMSE 0.11
0.28
0.31
0.023
0.35
0.24
A b
−0.074795 −0.015977 −0.004834 0.006962
0.018715
−0.001425
B b
0.080798
0.022342
0.003455
−0.010544 −1.819612 0.018197
R 2 b,c
0.990739
0.999063
0.999746
0.999714
0.998167
0.999912
σ b,d
0.08
0.025
0.015
0.017
0.19
0.19
B3LYP Mean 0.63
0.48
0.49
0.043
-.St. dev. 0.13
0.35
0.29
0.022
-.RMSE 0.65
0.59
0.57
0.048
0.53
0.47
A b
−0.093027 −0.023958 −0.015281 −0.003991 0.020715
−0.003057
B b
0.095014
0.028995
0.014730
0.000901
−1.993258 0.089102
R 2 b,c
0.988016
0.998736
0.999250
0.999539
0.992041
0.999644
σ b,d
0.09
0.029
0.025
0.021
0.40
0.39
Source Penocchio et al. (2015)
a Number of bonds
b Linear regression: A = 1 − slope; B = intercept
c Square of the correlation coefficient
d Standard deviation of the estimate
Mean ¯
e =
1
n
e i
RMSE
e 2
i
n
Std σ =
(e i −¯ e) 2
n−1
convenient to develop the PES in Taylor series as a function of the nuclear displacement coordinates around its minimum. In such a case, the first-order terms are zero.
The coefficients of this expansion are called the force field. They are usually divided
into two parts: the harmonic (or quadratic) force field which is the most important
term in the expansion and the anharmonic force field. The expansion is written in
the following form
V =
1
2
i j
f i j R i R j +
1
6
i jk
f i jk R i R j R k +
1
24
i jkl
f i jk R i R j R k R l + · · · (2.40)
where R denotes a set of nuclear displacement coordinates (internal coordinates,
Cartesian coordinates, …). Unfortunately, different systems of coordinates have to
be used and the transformation of the force constants to another coordinate system
2 Computational Methods
Table 2.18 Mean, standard deviation (St. dev.), root-mean-square error (RMSE) and standard
error of estimate (SEE) for the B2PLYP/VTZ and B3LYP/SNSD deviations from B2PLYP/VTZ
semiexperimental equilibrium structures (distances in pm and angles in degree)
CH
CC
CO
CN
∠(HCH)
∠(CCC)
n a
82
45
30
14
29
22
B2PLYP Mean 0.02
0.01
0.27
0.015
-.St. dev. 0.11
0.28
0.15
0.018
-.RMSE 0.11
0.28
0.31
0.023
0.35
0.24
A b
−0.074795 −0.015977 −0.004834 0.006962
0.018715
−0.001425
B b
0.080798
0.022342
0.003455
−0.010544 −1.819612 0.018197
R 2 b,c
0.990739
0.999063
0.999746
0.999714
0.998167
0.999912
σ b,d
0.08
0.025
0.015
0.017
0.19
0.19
B3LYP Mean 0.63
0.48
0.49
0.043
-.St. dev. 0.13
0.35
0.29
0.022
-.RMSE 0.65
0.59
0.57
0.048
0.53
0.47
A b
−0.093027 −0.023958 −0.015281 −0.003991 0.020715
−0.003057
B b
0.095014
0.028995
0.014730
0.000901
−1.993258 0.089102
R 2 b,c
0.988016
0.998736
0.999250
0.999539
0.992041
0.999644
σ b,d
0.09
0.029
0.025
0.021
0.40
0.39
Source Penocchio et al. (2015)
a Number of bonds
b Linear regression: A = 1 − slope; B = intercept
c Square of the correlation coefficient
d Standard deviation of the estimate
Mean ¯
e =
1
n
e i
RMSE
e 2
i
n
Std σ =
(e i −¯ e) 2
n−1
convenient to develop the PES in Taylor series as a function of the nuclear displacement coordinates around its minimum. In such a case, the first-order terms are zero.
The coefficients of this expansion are called the force field. They are usually divided
into two parts: the harmonic (or quadratic) force field which is the most important
term in the expansion and the anharmonic force field. The expansion is written in
the following form
V =
1
2
i j
f i j R i R j +
1
6
i jk
f i jk R i R j R k +
1
24
i jkl
f i jk R i R j R k R l + · · · (2.40)
where R denotes a set of nuclear displacement coordinates (internal coordinates,
Cartesian coordinates, …). Unfortunately, different systems of coordinates have to
be used and the transformation of the force constants to another coordinate system