Results

The BLYP functional is known to perform quite well on the G2 set [98,100,106], making it an ideal starting point for the optimizations. The results of modifying the BLYP parameters are listed in Table 4.1.

  

Table 4.1: Modified BLYP RMS deviations (kcal/mol)

Functional	RMS deviation
B(0.0042)-LYP	5.290
B(0.0035)-LYP	5.069
B(0.0042) + 1.0431LYP	4.963
B-LYP(optimum)	4.848

The unmodified BLYP functional (with Becke's $\beta$ parameter of 0.0042 in the exchange part) gives an RMS error of 5.290 kcal/mol. The corresponding mean absolute deviation is 4.11 kcal/mol. Next, the Becke $\beta$ parameter was optimized, moving to 0.0035, giving an RMS deviation of 5.069 kcal/mol. Third, an optimal linear combination of the original Becke and LYP parts is found, lowering the RMS deviation to 4.963 kcal/mol. A final, fuller optimization only lowers the RMS error to 4.848 kcal/mol. (This has a Dirac coefficient of 1.0072 times the original, a $\beta$ value of 0.003705 and LYP parameters of a,b,c,d = 0.049,0.108,0.24,0.342.)

The next step is to see the effect of the different $\beta$ values through external mixing. By taking a combination of B(0.0035)-LYP, B(0.0042)-LYP, Hartree-Fock-Slater and Hartree-Fock-Becke theories the RMS deviation is lowered to 4.543 kcal/mol. Clearly the linear combination of two Becke functionals is a better exchange functional than either separately. The external optimization coefficients are
$$\begin{align}9.558\mathrm{B(0.0035)LYP} - 7.727\mathrm{B(0.0042)LYP} - 0.861\mathrm{HFS} + 0.743\mathrm{HFB(0.0042)}. \end{align}$$
These coefficients were then refined via an internal optimization, along with a redetermination of the LYP parameters. The complete functional is then
$$\begin{align}\begin{split} E_{xc} &= 1.030952 E_{x}^{D30} + 10.4017 \Delta E_{x}... ....0042) \\ & \quad + E_{c}^{LYP}(0.055,0.158,0.25,0.3505) \end{split}\end{align}$$
where $\Delta E_{x}^{B88}$ represents the Becke correction, that is the B88 functional without the Dirac term. With this functional the RMS deviation is lowered to 4.237 kcal/mol and the mean absolute deviation is 3.215 kcal/mol. The non-LYP part of the above functional has been termed the double-Becke functional, while the complete functional is called `Empirical Density Functional 1' or EDF1.

  

Table 4.2: External linear mixing of Hartree-Fock (kcal/mol)

Functional Combination	RMS deviation
BLYP + HFB(0.0042) + HFS	4.920
BLYP + HFB(0.0042) + HFS + HF	4.499
EDF1 + HFB(0.0042) + HFB(0.0035) + HFS + BLYP	4.211
EDF1 + HFB(0.0042) + HFB(0.0035) + HFS + BLYP + HF	4.211

The third step is to investigate the effect of adding a fraction of Fock exchange to EDF1. Table 4.2 gives the results of external mixing of Hartree-Fock with BLYP and EDF1 plus their components. Clearly, there is strong HF mixing with BLYP, but virtually none with EDF1. This is confirmed by trying to mix in HF with an internal optimization of the EDF1 components and Fock exchange. The HF coefficient is less than 0.001 and there is no significant lowering of the RMS error.

To asses the usefulness of EDF1 as a density functional it has been used to obtain the thermochemistry of the molecules in the G2 set, using the 6-31G* basis. The results are listed in Table 4.3 and Table 4.4. In addition, the BLYP and B3LYP functionals have been included for comparison. Perhaps surprisingly, B3LYP performs quite poorly, being inferior to both BLYP and EDF1. However, the parametization of the B3LYP functional was carried out using what is, in effect, an infinite basis. It is therefore not unreasonable to expect performance to improve if it was reoptimized for the 6-31G* basis. An internal reoptimization along these lines shows a decrease in the Fock exchange coefficient to about 5%, with an RMS error of 4.65 kcal/mol, still considerably inferior to EDF1.

  

Table 4.3: Deviations from experiment for various functionals

	Exp.	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
atomization energies (kcal/mol)
H2	103.3	-0.1	-0.6	-3.1
LiH	56.0	1.0	1.0	1.3
BeH	46.9	-7.0	-7.9	-7.2
CH	79.9	-0.3	0.3	-0.4
CH2 (3B1)	179.6	1.9	0.1	-2.3
CH2 (1A1)	170.6	4.0	3.8	2.2
CH3	289.2	1.5	-0.8	-3.7
CH4	392.5	3.9	0.6	-3.7
NH	79.0	-3.5	-1.6	-2.7
NH2	170.0	-2.6	-0.2	-2.6
NH3	276.7	2.6	4.4	-0.2
OH	101.3	0.7	2.8	0.4
OH2	219.3	7.6	10.3	4.6
FH	135.2	5.3	7.7	2.9
SiH2(1A1)	144.4	1.5	0.2	-0.6
SiH2(3B1)	144.4	1.5	0.2	-0.6
SiH3	214.0	4.4	1.0	-0.3
SiH4	302.8	6.0	1.0	0.7
PH2	144.7	-0.7	-1.0	-2.8
PH3	227.4	4.4	3.0	0.4
SH2	173.2	7.5	7.1	3.4
ClH	102.2	6.5	6.4	3.5
Li2	24.0	4.2	4.2	5.4
LiF	137.6	2.3	5.8	4.4
HCCH	388.9	8.4	11.3	4.1
H2CCH2	531.9	7.0	5.6	-1.3
H3CCH3	666.3	9.5	3.7	-3.0
CN	176.6	-3.7	7.8	-2.1
HCN	301.8	-2.1	6.6	-1.5

Table 4.3 (continued)

	Exp.	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
CO	256.2	2.5	10.5	2.4
HCO	270.3	-4.4	3.6	-6.3
H2CO	357.2	-1.0	4.9	-4.5
H3COH	480.8	6.9	7.8	-0.1
N2	225.1	-4.5	8.8	0.2
H2NNH2	405.4	-0.1	5.2	-2.4
NO	150.1	-8.5	3.6	-6.7
O2	118.0	-13.1	1.3	-13.9
HOOH	252.3	0.5	11.3	-0.2
F2	36.9	-9.1	3.8	-7.3
CO2	381.9	-2.8	11.9	-5.4
Na2	16.6	-0.7	-0.1	1.3
Si2	74.0	1.9	9.0	0.8
P2	116.1	2.5	10.4	3.7
S2	100.7	1.8	7.4	0.0
Cl2	57.2	7.7	10.9	6.6
NaCl	97.5	7.4	6.7	6.7
SiO	190.5	3.6	12.7	6.2
SC	169.5	3.6	10.8	2.6
SO	123.5	-1.1	9.0	-1.4
ClO	63.3	-3.5	5.8	-2.9
ClF	60.3	-0.9	6.1	-0.7
CH3Cl	371.0	7.1	5.1	-0.2
Si2H6	500.1	13.2	4.5	4.1
CH3SH	445.1	11.1	8.4	2.1
HOCl	156.3	4.1	10.8	3.0
SO2	254.0	16.3	35.9	15.3
ionization potentials (V)
H	13.60	0.12	0.08	0.06
He	24.59	-0.12	-0.19	-0.20

Table 4.3 (continued)

	Exp.	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
Li	5.39	-0.13	-0.15	-0.12
Be	9.32	0.33	0.29	0.36
B	8.30	-0.27	-0.30	-0.26
C	11.26	-0.13	-0.19	-0.18
N	14.54	-0.01	-0.09	-0.13
O	13.61	-0.54	-0.43	-0.31
F	17.42	-0.33	-0.26	-0.24
Ne	21.56	-0.22	-0.17	-0.28
Na	5.14	-0.19	-0.19	-0.07
Mg	7.65	0.02	0.00	0.14
Al	5.98	0.11	0.04	0.04
Si	8.15	0.20	0.11	0.12
P	10.49	0.29	0.17	0.16
S	10.36	-0.01	-0.05	0.01
Cl	12.97	0.07	0.00	0.04
Ar	15.76	0.11	0.01	0.00
CH4	12.62	0.13	0.00	0.07
NH3	10.18	0.19	0.20	0.13
OH	13.01	-0.11	-0.06	-0.04
OH2	12.62	0.17	0.19	0.11
FH	16.04	0.08	0.11	0.00
SiH4	11.00	0.21	0.02	0.18
PH	10.15	0.15	0.04	0.03
PH2	9.82	0.05	-0.05	-0.05
PH3	9.87	0.18	0.14	0.17
SH	10.37	0.09	0.03	0.08
SH2	10.47	0.24	0.16	0.18
ClH	12.75	0.19	0.10	0.10
HCCH	11.40	0.40	0.36	0.32
H2CCH2	10.51	0.35	0.35	0.28

Table 4.3 (continued)

	Exp.	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
CO	14.01	0.01	-0.12	0.07
N2	15.58	0.24	-0.15	0.21
O2	12.07	-0.47	-0.79	-0.53
P2	10.53	0.32	-0.36	0.20
S2	9.36	-0.02	-0.26	-0.13
Cl2	11.50	0.30	0.04	0.23
ClF	12.66	0.19	-0.02	0.16
SC	11.33	-0.06	-0.14	-0.05
electron affinities (eV)
C	1.26	-0.07	-0.02	-0.06
CH	1.24	-0.07	-0.02	-0.07
CH2	0.65	-0.07	0.05	0.05
CH3	0.08	0.14	0.24	0.21
CN	3.82	-0.06	-0.18	0.02
NH	0.38	-0.07	0.09	0.07
NH2	0.74	0.08	0.21	0.15
NO	0.02	-0.37	-0.36	-0.30
O	1.46	-0.26	-0.06	-0.11
OH	1.83	0.02	0.19	0.08
O2	0.44	-0.14	-0.11	0.07
F	3.40	-0.22	-0.02	-0.17
Si	1.38	0.19	0.13	0.18
SiH	1.28	0.14	0.09	0.13
SiH2	1.12	0.08	0.04	0.08
SiH3	1.44	0.11	0.11	0.17
P	0.75	-0.10	-0.07	-0.02
PH	1.00	0.01	0.02	0.08
PH2	1.26	0.12	0.11	0.16
PO	1.09	-0.05	-0.17	-0.06
S	2.08	-0.04	-0.03	0.02

Table 4.3 (continued)

	Exp.	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
SH	2.31	0.07	0.07	0.09
S2	1.66	0.07	-0.03	0.12
Cl	3.62	0.01	-0.01	0.00
Cl2	2.39	-0.69	-0.70	-0.56
proton affinities (kcal/mol)
H2	100.8	10.7	11.5	8.5
HCHH	152.3	0.5	-0.2	-2.5
NH3	202.5	1.1	-0.2	-1.7
H2O	165.1	5.3	4.1	2.9
SiH4	154.0	4.0	5.7	2.9
PH3	187.1	4.0	2.8	1.5
H2S	168.8	3.4	3.5	1.0
HCl	133.6	5.6	6.5	3.4

  

Table 4.4: RMS errors for functionals

RMS Errors (kcal/mol)	Exp.-BLYP	Exp.-B3LYP	Exp.-EDF1
atomization energies	5.75	8.10	4.41
ionization potentials	5.14	5.05	4.34
electron affinities	4.39	4.40	3.79
proton affinities	5.23	5.51	3.77

The overall improvement in moving from B3LYP (or BLYP) to EDF1 is largely due to the better atomization energies and proton affinities. The electron addition and removal energies are only slightly superior. It is interesting that the worst EDF1 results (atomization energies of SO₂ and O₂, ionization energy of O₂, electron affinity of Cl₂, and proton affinity of H₂) are also problematic cases for BLYP and B3LYP. This confirms the underlying similarity of each of the three functionals, that they are all made from essentially the same main components.

By writing density functionals in the form
$$\begin{align}E_{xc}[\rho] = \int \rho^{4/3}({\bf r}) g(x) \, d{\bf r} \end{align}$$
the double-Becke functional can be compared with the original B88 form. The B88 g(x) is
$$\begin{align}g_{B88}(\beta,x) = C_{0} - \frac{\beta x^{2}}{1+6\beta x \sinh^{-1}(x)} \end{align}$$
with $\beta = 0.0042$ and C₀ is the coefficient of the Dirac functional. The new g(x) is
$$\begin{align}g_{doubleB}(x) = -0.922818 C_{0} + 10.4017 g_{B88}(0.0035,x) - 8.44793 g_{B88}(0.0042,x) \end{align}$$
These two functions are plotted in Figure 4.1.

  

Figure 4.1: The g(x) functions for the B88 and double-Becke exchange functionals

xx LegA g_B88(x) LegB g_doubleB(x)

At x=0 the double-Becke value is slightly below that of the uniform electron gas. The double-Becke curve is also much flatter at the origin. This can be seen from the initial term in the Taylor expansion
$$\begin{align}g(x) = g(0) + \frac{1}{2}g''(0)x^{2}+ \ldots \end{align}$$
which is g''(0) = -0.00184, compared with -0.0084 from the original B88 form. The double-Becke value is much closer to the Sham-Kleinman value of -0.00387 . Also, the double-Becke curve is below B88 until x=4 and the curves cross again at x=9.7 .