Molecular Results

To begin, the effect of CASE on the molecular orbitals of H₂O is examined. The MP2/6-31G* geometry is used. The HF/6-31G* calculations using $\omega = 0$ and $\omega = 0.1$ give total energies of -76.00981 and -71.72696 respectively. Thus, as expected, there is a huge effect on the total energy. The MO energies are presented in Figure 6.2. This figure reveals the interesting effect that Coulomb attenuation raises the occupied MO energies, but has very little effect on the unoccupied MOs.

**Figure:** Molecular orbital energies (a.u.) for H₂O using CASE HF/6-31G* (core MO not shown)
LegA $\omega = 0$ LegB $\omega = 0.1$ $\includegraphics[scale=0.8]{plot2.epsf}$

**Table:** MO coefficients and energies for the HOMO and LUMO of H₂O using HF/6-31G*
	HOMO (B₁)		LUMO (A₁)
	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$
O (1s)	0	0	0.10002	0.09992
O (2s)	0	0	-0.05859	-0.05873
O (2 p_x)	0	0	0	0
O (2 p_y)	0.63998	0.63962	0	0
O (2 p_z)	0	0	-0.22243	-0.22247
O (3s)	0	0	-1.38818	-1.38818
O (3 p_x)	0	0	0	0
O (3 p_y)	0.51110	0.51148	0	0
O (3 p_z)	0	0	-0.51217	-0.51233
O (3 d_xx)	0	0	0.05532	0.05519
O (3 d_xy)	0	0	0	0
O (3 d_yy)	0	0	0.07027	0.07016
O (3 d_xz)	0	0	0	0
O (3 d_yz)	0.03347	0.03334	0	0
O (3 d_zz)	0	0	0.04191	0.04180
H₁ (1s)	0	0	0.05745	0.05744
H₁ (2s)	0	0	1.03009	1.02992
H₂ (1s)	0	0	0.05745	0.05744
H₂ (2s)	0	0	1.03009	1.02992
MO energy	-0.49736	-0.38174	+0.20821	+0.21353

The MO coefficients of the highest-occupied (HOMO) and lowest-unoccupied (LUMO) molecular orbitals and their energies are listed in Table 6.2. These MOs typify the changes found for all the molecular orbitals on introducing attenuation. It is clear that the wavefunction is little affected by a small attenuation.

**Table:** UHF/6-31G** energies (hartrees) of H₂ as a function of bond length (Å)
	E(R)		$E(R)-E(\infty)$
R	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$
0.4	-0.93620	-0.71184	+0.06027	+0.06107
0.5	-1.06148	-0.83730	-0.06502	-0.06439
0.6	-1.11393	-0.88993	-0.11747	-0.11702
0.7	-1.13050	-0.90667	-0.13403	-0.13376
0.8	-1.12843	-0.90479	-0.13197	-0.13188
0.9	-1.11652	-0.89305	-0.12005	-0.12014
1.0	-1.09947	-0.87619	-0.10301	-0.10329
1.1	-1.07994	-0.85685	-0.08348	-0.08394
1.2	-1.05940	-0.83649	-0.06293	-0.06359
1.3	-1.04147	-0.81847	-0.04501	-0.04556
1.4	-1.02864	-0.80553	-0.03217	-0.03262
1.5	-1.01950	-0.79631	-0.02303	-0.02340
1.6	-1.01299	-0.78974	-0.01653	-0.01683
1.7	-1.00836	-0.78505	-0.01189	-0.01214
1.8	-1.00505	-0.78169	-0.00858	-0.00878
1.9	-1.00267	-0.77928	-0.00621	-0.00637
2.0	-1.00097	-0.77754	-0.00450	-0.00464
$\infty$	-0.99647	-0.77291	-0.00000	-0.00000

Given the very large changes to the total energies for the hydrogen atom and the water molecule, it might seem unlikely that bond dissociation energies could be reproduced well. To test this, the UHF/6-31G** potential curve of H₂ was scanned with and without Coulomb attenuation. The total energies are listed in Table 6.3. Despite the fact that the total energies are 224 millihartrees higher than their $\omega = 0$ counterparts, the difference is so constant over a wide range of bond distances that the spectroscopic parameters remain effectively unchanged. For example the equilibrium bond distance, r_e, increases from 0.7326 to 0.7338 Å, D_e falls from 354.1 to 353.5 kJ mol^-1 and $\nu_{e}$ falls from 4635 to 4616 cm^-1. For this simple case at least, the neglect of the background has meant no deterioration in bond dissociation.

**Table:** HF/6-31G* total energies (hartrees), atomization energies (kJ mol^-1 and MP2/6-31G* correlation energies (millihartrees) of various molecules
	Total Energy		Atomiz. Energy		Correlation Energy
Molecule	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$
H₂	-1.12679	-0.90305	342.2	341.7	-17.4	-17.3
H₂CCH₂	-78.03107	-72.85255	1775.9	1776.1	-263.2	-261.7
H₂CO	-113.86372	-107.21602	1056.7	1058.9	-311.2	-309.9
H₂NNH₂	-111.16800	-104.40912	1061.2	1062.4	-336.4	-335.2
H₃CCH₃	-79.22854	-73.82606	2303.7	2302.8	-275.4	-274.0
H₃COH	-115.03419	-108.16217	1513.5	1513.7	-319.1	-318.0
HCCH	-76.81560	-71.86094	1200.9	1201.9	-260.6	-258.9
HCN	-92.87019	-87.23785	802.5	805.9	-296.8	-295.0
HCO	-113.24518	-106.70926	740.8	743.0	-295.2	-293.9
HOOH	-150.76012	-142.41897	514.0	516.4	-374.8	-373.7
Li₂	-14.86689	-13.52642	10.9	13.6	-20.0	-18.3
LiF	-106.93418	-101.18581	361.9	355.4	-195.3	-194.9
LiH	-7.98087	-7.19726	134.6	131.7	-15.6	-15.4
BeH	-15.14731	-13.91139	215.6	211.2	-24.1	-23.9
CH	-38.26485	-35.78739	225.1	225.3	-77.6	-77.2
CH₂ (¹A₁)	-38.87219	-36.28291	511.6	511.7	-101.8	-101.4
CH₂ (³B₁)	-38.92142	-36.33159	640.9	639.5	-86.0	-85.7
CH₃	-39.55892	-36.85718	1006.5	1004.8	-114.1	-113.6
CH₄	-40.19507	-37.38157	1368.6	1366.9	-142.0	-141.3
CN	-92.17398	-86.65255	282.7	283.8	-220.8	-219.7
CO	-112.73448	-106.31014	708.1	709.7	-293.7	-292.4
CO₂	-187.62841	-177.14614	996.9	1002.0	-490.0	-487.9
N₂	-108.93540	-102.62432	431.9	434.9	-326.2	-324.4
NH	-54.95924	-51.69139	198.4	198.5	-102.2	-101.9
NH₂	-55.55731	-52.17765	460.5	460.5	-136.4	-136.0
NH₃	-56.18384	-52.69221	797.4	796.9	-173.5	-173.0
NO	-129.24730	-122.03288	204.6	207.0	-317.2	-315.9
O₂	-149.60681	-141.48969	102.3	105.8	-347.5	-346.2

Table 6.4 (continued)

	Total Energy		Atomiz. Energy		Correlation Energy
Molecule	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$
OH	-75.38186	-71.21088	261.7	261.8	-141.3	-141.0
OH₂	-76.00981	-71.72696	602.3	602.2	-189.4	-189.0
F₂	-198.67283	-188.52364	-149.9	-146.9	-366.0	-365.1
FH	-100.00229	-94.81532	365.2	365.1	-181.9	-181.5

With the realization that attenuation has little effect on the bonding of H₂ the next step is to test CASE on a range of more complicated electronic structures. Table 6.4 presents the atomization energies for 32 first and second row molecules. Zero-point vibrational corrections are not included. Again, the total energies are raised by large amounts. But there is almost imperceptible movement in the relative (that is, atomization) energies which are typically 0-3 kJ mol^-1. The largest error in the set (6.4 kJ mol^-1) occurs for LiF where the attenuated calculation fails to capture all of the Coulombic stabilization. This is the most ionic species of the 32 molecules, so it is not surprising that CASE `fails' here as the non-polar assumption is least valid.

While CASE was designed with Hartree-Fock in mind, given the good cancellation of error, it may be possible to obtain useful correlation energies with the CASE integrals. The simplest correlation method to try this with is second-order Møller-Plesset perturbation theory (equation (1.75)). To investigate the effect of Coulomb attenuation on the integrals required, orbital energies from an $\omega = 0$ Hartree-Fock calculation have been used. The MP2/6-31G* correlation energy has then been calculated for each of the 32 molecules above, and is presented in the last two columns of Table 6.4. The numerators of equation (1.75) are the difference between two integrals which allows a cancellation of error, producing good agreement with unattenuated calculations. The attenuated correlation calculations are systematically lower than traditional values, but the difference is only slight in all cases. It seems that neglect of the background is of little importance in the calculation of correlation energies.

With the use of localized orbitals [194] and a Laplace transform [195,196] to remove the denominators of equation (1.75), the introduction of CASE here may provide a way to lower the scaling of MP2; however, this has not been investigated here.

**Table 6.5:** UHF/6-31G* total energies (hartrees) and ionization energies (eV)
	Total Energy (atom)		Total Energy (cation)		Ionization Energy
Atom	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$	$\omega = 0$	$\omega = 0.1$	$\Delta$
H	-0.49823	-0.38645	-0.00000	-0.00000	13.56	10.52	3.04
He	-2.85516	-2.51752	-1.99362	-1.76853	23.44	20.38	3.06
Li	-7.43137	-6.76063	-7.23554	-6.67201	5.33	2.41	2.92
Be	-14.56694	-13.44452	-14.27552	-13.26483	7.93	4.89	3.04
B	-24.52204	-22.83410	-24.23406	-22.66026	7.84	4.73	3.11
C	-37.68086	-35.31512	-37.28708	-35.03592	10.72	7.60	3.12
N	-54.38544	-51.22933	-53.87220	-50.83083	13.97	10.84	3.12
O	-74.78393	-70.72469	-74.34264	-70.39817	12.01	12.46	3.13
F	-99.36496	-94.28980	-98.79206	-93.83177	15.59	12.46	3.13
Ne	-128.47441	-122.27059	-127.75171	-121.66270	19.67	16.54	3.12

There are some properties for which the CASE approximation performs poorly, though. The most severe failure is for atomic ionization energies, which are listed in Table 6.5. The UHF/6-31G* ionization energies show a marked decrease with the introduction of attenuation. The background is important here because, for the first time in this chapter, there is a distant interaction that does not cancel -- an electron is lost from the system, which interacts with the cation that remains.

A simple estimate for this effect can be computed by calculating the effect of the first correction for the background on ionization. Taking just the first term of the Taylor expansion to represent the background,
$\begin{align}\frac{\erf(\omega r)}{r} \approx \frac{2 \omega}{\sqrt{\pi}}. \end{align}$
Then the effect of the background on the ionization energy is that of the departing electron interacting (via the background operator) with the charge that remains:
$\begin{align}I_{L} &= \langle \rho_{cation} \vert \frac{2 \omega}{\sqrt{\pi}} \v... ...n} \vert \rho_{electron}\rangle \\ &= \frac{2 \omega}{\sqrt{\pi}}, \end{align}$
which has a value of 3.12 eV for $\omega = 0.1$ , in very good agreement with the values listed in Table 6.5.

Note that this will not be the case for electron affinities, as the molecule that interacts with the approaching electron is neutral, rather than charged.