Minimal set of moleculeadapted atomic orbitals from maximum overlap criterion
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Abstract
The criterion of maximum overlap with the canonical freeatom orbitals is used to construct a minimal set of moleculeintrinsic orthogonal atomic orbitals that resemble the most their promolecular origins. Partial atomic charges derived from population analysis within representation of such moleculeadopted atomic orbitals are examined on example of firstrow hydrides and compared with charges from other methods. The maximum overlap criterion is also utilized to approximate the exact freeatom orbitals obtained from ab initio calculations in any arbitrary basis set and the influence of the resulting fitted canonical atomic orbitals on properties of moleculeadopted atomic orbitals is briefly discussed.
Keywords
Minimal basis set Atomic orbitals Maximum overlap criterion Partial atomic charges Moleculeintrinsic orbitals1 Introduction
It has recently been argued [1] within the framework the Orbital Communication Theory (OCT) [2, 3, 4] of the chemical bond that the minimum basis (MB) of atomic orbitals (AO) occupied in the promolecular system of noninteracting atoms gives rise to the most intuitively ”chemical” account of the bond covalency/ionicity and gives understanding of diverse factors conditioning the efficiency of the AO interactions. It has turned out that, in the case of molecules with typical covalent bonds, amount of information about electron localization calculated within the MBset of atomic orbitals is (to a certain degree) invariant with respect to basisset enlargement [1]. The maximum overlap criterion (MOC) [5, 6] has been proposed to numerically confirm that the MOCapproximated (fitted) small basis and the reference minimum set of AOs indeed generate practically identical OCTindices.
The criterion of maximum overlap has also been used [7] to approximate the wave function calculated within a general (extended) basis set by the molecular orbitals (MO) calculated within the external minimal basis (EMB), i.e. Huzinagas MINI basis functions [8]. It has been demonstrated that the electron population analysis (EPA) based on the resulting EMBorbitals gives rise to definitely more reliable and sensible partial atomic charges (PICs) than the corresponding Mulliken’s and Löwdins population analyses (MPA [9, 10, 11, 12] and LPA [13], respectively), especially if calculated within extended sets of basis functions. Moreover, EMBcharges have turned out to be comparable to PACs calculated within representation of natural atomic orbitals (NAO) [14], as well as preserve the proper convergence profile when one systematically enlarges the basis set.
The main goal of this paper is to utilize the MOCscheme (in the form presented in [7]) to generate the moleculeintrinsic minimalbasis (IMB) orbitals that resembles the most (in the leastsquares sense) the canonical freeatom orbitals (FAO) of the relevant system promolecule.
2 Method details
 1.Evaluating the missing \(n_\chi \times {n^v_\mathrm{v }}\) LCmatrix \(\bar{\mathbf{C}}^{v}_\chi \) from the virtual MOsubspace \(\mathbf C ^v_\chi \) as follows:where the unitary matrix \(\mathbf U ^{v}_\varphi \) diagonalize the appropriate metric matrix \(\mathbf W _{\varphi }^{v}\),$$\begin{aligned} \bar{\mathbf{C}}^{v}_\chi = \mathbf C ^{v}_\chi \left( \mathbf U ^{v}_\varphi \right) _{1\rightarrow n^v_\mathrm{v }}, \end{aligned}$$(5)More details about the virtual valence MOsubspace given by \(\bar{\mathbf{C}}^{v}_\chi \) one can find in the original work [15]. The subspaces of occupied and virtual valenceMOs can be collectively written as \({\bar{\psi }}\rangle \) and the corresponding \(n_\chi \times {b}\) matrix \(\bar{\mathbf{C}}_{\chi }=(\mathbf C ^o_{\chi } \bar{\mathbf{C}}^v_{\chi })\).$$\begin{aligned} \mathbf W _{\varphi }^{v} = \mathbf V _{\varphi }^{v\dagger }\mathbf V _{\varphi }^{v} = \mathbf U ^{v}_\varphi \mathbf w ^{v}_\varphi \mathbf U ^{v\dagger }_\varphi , \quad \text {and}\quad \mathbf V _{\varphi }^{v} = \mathbf B ^{o\dagger }_{\chi }\mathbf S _{\chi }\mathbf C ^v_{\chi }. \end{aligned}$$(6)
 2.Using the maximum overlap criterion,to find the moleculeintrinsic orthogonal atomic orbitals \(\varphi \rangle \) that resemble the most canonical atomic orbitals \(\varphi ^0\rangle \). Then, according to [7] we get:$$\begin{aligned} \text {tr}\langle \varphi ^0\varphi \rangle = \text {maximum}, \end{aligned}$$(7)where$$\begin{aligned} \varphi \rangle = \chi \rangle \mathbf B _{\chi },\quad \mathbf B _{\chi } = \mathbf T \left( \mathbf T ^{\dagger }\mathbf T \right) ^{1/2}, \end{aligned}$$(8)$$\begin{aligned} \mathbf T = \mathbf {B} ^{0\dagger }_{\chi }\mathbf S _{\chi }\bar{\mathbf{C}}_{\chi }. \end{aligned}$$(9)
3 Numerical results
Numerical results presented in this section were obtained using the special program written by authors to perform MOCcalculations as well as the ab initio quantum chemistry package GAMESS [17], the Natural Bond Orbitals software NBO 5.0 [18] and the molecular visualization program MOLDEN [19]. Molecular calculations were carried out for species diversified by means of chemical bond character, i.e. the following firstrow hydrides: LiH, BeH\(_2\), BH\(_3\), CH\(_4\), NH\(_3\), H\(_2\)O and HF (for comparison calculations for LiF were performed as well) at the RHF theory level and using experimental geometries whereas calculations for free atoms were performed using the ROHF/GVB method with explicitly assumed fractional occupation numbers to assure sphericall symmetry of atoms (more details can be found in ”Further Information” section of GAMESS documentation [17]).
3.1 IMBorbitals from MOC

MullikenPA (\(Q^\mathrm{MPA }_\mathrm X \)) [9, 10, 11, 12],

LöwdinPA (\(Q^\mathrm{LPA }_\mathrm X \)) [13],

NaturalPA (\(Q^\mathrm{NPA }_\mathrm X \)) [14],

electrostatic potential (ESP)derived atomic charges (\(Q^\mathrm{ESP }_\mathrm X \)) [20],

the recently proposed population analysis involving the EMB of AOs from the criterion of maximum overlap with the Huzinagas MINI basis functions, (\(Q^\mathrm{EMB }_\mathrm X \)) [7].
Partial atomic charges of central atoms of firstrow hydrides and LiF molecule calculated at their experimental geometries and using various populationanalysis schemes
Molecule  \(Q^\mathrm{MPA }_\mathrm{X}\)  \(Q^{\mathrm{LPA}}_\mathrm{X}\)  \(Q^\mathrm{NPA}_\mathrm{X}\)  \(Q^{\mathrm{ESP}}_\mathrm{X}\)  \(Q^{\mathrm{EMB}}_\mathrm{X}\)  \(Q^{\mathrm{IMB}}_\mathrm{X}\) 

LiH  0.4372  0.1938  0.8468  0.8444  0.6315  0.6226 
BeH\(_2\)  0.4766  0.1004  1.2289  0.7282  1.0475  1.2172 
BH\(_3\)  0.1025  0.0415  0.4192  0.6295  0.0062  0.0452 
CH\(_4\)  0.0481  0.0203  0.7255  0.6015  0.7099  0.5660 
NH\(_3\)  0.4729  0.0403  1.0342  0.9266  1.0978  0.7883 
H\(_2\)O  0.5704  0.0004  0.9276  0.7379  1.0603  0.7663 
HF  0.4080  0.0510  0.5609  0.4565  0.6680  0.5000 
LiF  0.7547  0.4594  0.9781  0.8849  0.9932  0.9450 
For the most part atomic charges \(Q^\mathrm{NPA }_\mathrm X \), \(Q^\mathrm{ESP }_\mathrm X \), \(Q^\mathrm{EMB }_\mathrm X \) and \(Q^\mathrm{IMB }_\mathrm X \) sharply differ from MPA and LPAderived ones; they converge to definitely more reasonable values and thus they seem to display adequate picture of molecular electronic structure in atomic resolution. This is evident especially if one compares results for LiF; only atomic charges from NPA, ESP, EMB and IMB correctly predict predominatingly ionic (90–100 %) character of the chemical bond Li–F. However, as follows from a thorough analysis of results from Table 1, in most cases partial atomic charges \(Q^\mathrm{NPA }_\mathrm X \) and \(Q^\mathrm{EMB }_\mathrm X \) reveal quite good correlation and both tend to predict slightly more polarized bond densities than \(Q^\mathrm{IMB }_\mathrm X \) and \(Q^\mathrm{ESP }_\mathrm X \).
 \(\overline{\Delta N}_\mu ^{\text {IMB/EMB}}\quad =\quad 0.0145e \text {(1s)},\quad 0.2590e \text {(2s)}, \quad 0.0412e \text {(2p)}\),Table 2
Electron populations of IMB and EMBorbitals as well as the minimalbasis subset of NAOs for central atoms of firstrow hydrides calculated at their experimental geometries
Molecule
\(N^\mathrm{IMB }_\mathrm{1s }\)
\(N^\mathrm{IMB }_\mathrm{2s }\)
\(N^\mathrm{IMB }_\mathrm{2p_x }\)
\(N^\mathrm{IMB }_\mathrm{2p_y }\)
\(N^\mathrm{IMB }_\mathrm{2p_z }\)
LiH
2.0000
0.3774
0.0000
0.0000
0.0000
BeH\(_2\)
2.0000
0.7828
0.0000
0.0000
0.0000
BH\(_3\)
2.0000
0.9705
0.9922
0.9922
0.0000
CH\(_4\)
2.0000
1.1119
1.1514
1.1514
1.1514
NH\(_3\)
2.0000
1.4134
1.2641
1.2641
1.8467
H\(_2\)O
2.0000
1.6502
2.0000
1.3850
1.7312
HF
2.0000
1.8365
2.0000
2.0000
1.6635
Molecule
\(N^\mathrm{EMB }_\mathrm{1s }\)
\(N^\mathrm{EMB }_\mathrm{2s }\)
\(N^\mathrm{EMB }_\mathrm{2p_x }\)
\(N^\mathrm{EMB }_\mathrm{2p_y }\)
\(N^\mathrm{EMB }_\mathrm{2p_z }\)
LiH
1.9794
0.3891
0.0000
0.0000
0.0000
BeH\(_2\)
1.9796
0.9729
0.0000
0.0000
0.0000
BH\(_3\)
1.9731
1.2225
0.9054
0.9054
0.0000
CH\(_4\)
1.9790
1.5548
1.0587
1.0587
1.0587
NH\(_3\)
1.9919
1.8672
1.2069
1.2069
1.8250
H\(_2\)O
1.9969
1.9601
2.0000
1.3714
1.7319
HF
1.9988
1.9893
2.0000
2.0000
1.6798
Molecule
\(N^\mathrm{NPA }_\mathrm{1s }\)
\(N^\mathrm{NPA }_\mathrm{2s }\)
\(N^\mathrm{NPA }_\mathrm{2p_x }\)
\(N^\mathrm{NPA }_\mathrm{2p_y }\)
\(N^\mathrm{NPA }_\mathrm{2p_z }\)
LiH
1.9995
0.1324
0.0000
0.0000
0.0000
BeH\(_2\)
1.9998
0.6709
0.0000
0.0000
0.0000
BH\(_3\)
1.9997
0.9549
0.8085
0.8085
0.0000
CH\(_4\)
1.9997
1.1545
1.1857
1.1857
1.1857
NH\(_3\)
1.9997
1.4960
1.3358
1.3358
1.8398
H\(_2\)O
1.9999
1.7434
1.9925
1.4531
1.7168
HF
2.0000
1.9041
1.9963
1.9963
1.6536

\(\overline{\Delta N}_\mu ^{\text {NPA/IMB}}\quad =\quad 0.0002e \text {(1s)},\quad 0.0941e \text {(2s)}, \quad 0.0485e \text {(2p)}\),

\(\overline{\Delta N}_\mu ^{\text {NPA/EMB}}\quad =\quad 0.0142e \text {(1s)},\quad 0.2714e \text {(2s)}, \quad 0.0657e \text {(2p)}\).
3.2 MOCapproximated FAOs
Average (over all basis sets \(\mathfrak a \)–\(\mathfrak g \)) overlap integrals between MOCapproximated freeatom orbitals and the corresponding canonical FAOs obtained from ab initio calculations using reference basis sets \(\mathfrak d \) and \(\mathfrak g \)
Atom  \(\bar{\mathcal{S }}^\mathfrak{g }\)  \(\bar{\mathcal{S }}^\mathfrak{d }\)  \(\bar{\mathcal{S }}^\mathfrak{d }_\mathfrak{a \mathfrak c }\)  \(\bar{\mathcal{S }}^\mathfrak{d }_\mathfrak{e \mathfrak g }\)  \(\bar{\mathcal{S }}^\mathfrak{a }\) 

H  1.00000  1.00000  1.00000  0.99999  0.98295 
Li  0.99995  0.99995  0.99991  0.99998  0.98907 
Be  0.99995  0.99994  0.99991  0.99997  0.99140 
B  0.99994  0.99982  0.99993  0.99971  0.97287 
C  1.00000  0.99991  0.99999  0.99984  0.98912 
N  0.99999  0.99995  1.00000  0.99989  0.99438 
O  0.99999  0.99992  1.00000  0.99984  0.99306 
F  0.99999  0.99993  1.00000  0.99986  0.99276 
Atomaveraged  0.99998  0.99993  0.99997  0.99988  0.98820 
In accordance to expectations, results from Table 3 clearly indicate that using the maximum overlap criterion with the reference FAOs calculated within set of functions \(\mathfrak g \) allows one to reproduce canonical orbitals for freeatoms within any basis set \(\mathfrak a \)–\(\mathfrak f \) with the highest accuracy; typical deviation of MOCderived orbitals from ab initio FAOs is \(10^{5}\). Using the reference basisset with only one set of polarization funtions and without diffuse ones, \(\mathfrak d \), gives rise to very similar results provided that \(\chi ^{\prime x}\rangle =\mathfrak a,b,c \); approximation of more accurate FAOs (\(\chi ^{\prime x}\rangle =\mathfrak e,f,g \)) leads to slightly worse results, with deviation assuming \(10^{4}\). Obviously, for reference FAOs calculated with very poor set o functions the MOCprocedure does not allow one to obtain atomic orbitals of satisfying quality (atomaveraged deviation is up to \(10^{2}\)).
Comparison of partial atomic charges calculated within representation of exact IMBorbitals and using various basis sets with the relevant PACs calculated within representation of IMBorbitals involving MOCfitted FAOs from reference basis sets \(\mathfrak d \) and \(\mathfrak g \)
Molecule  PIC  \(\mathfrak a \)  \(\mathfrak b \)  \(\mathfrak c \)  \(\mathfrak d \)  \(\mathfrak e \)  \(\mathfrak f \)  \(\mathfrak g \) 

LiH  \(Q^\mathrm{IMB }_\mathrm{Li }\)  0.4764  0.5797  0.6185  0.6190  0.6216  0.6227  0.6226 
\(\mathcal Q _\mathrm{Li }^\mathfrak{g }\)  0.4764  0.5847  0.6189  0.6193  0.6213  0.6225  0.6226  
\(\mathcal Q _\mathrm{Li }^\mathfrak{d }\)  0.4764  0.5848  0.6188  0.6190  0.6190  0.6194  0.6206  
BeH\(_2\)  \(Q^\mathrm{IMB }_\mathrm{Be }\)  1.0223  1.1754  1.1998  1.2126  1.2154  1.2168  1.2172 
\(\mathcal Q _\mathrm{Be }^\mathfrak{g }\)  1.0223  1.1823  1.2016  1.2144  1.2153  1.2169  1.2172  
\(\mathcal Q _\mathrm{Be }^\mathfrak{d }\)  1.0223  1.1821  1.2001  1.2126  1.2125  1.2141  1.2145  
BH\(_3\)  \(Q^\mathrm{IMB }_\mathrm{B }\)  0.1491  0.0013  0.0034  0.0301  0.0407  0.0435  0.0452 
\(\mathcal Q _\mathrm{B }^\mathfrak{g }\)  0.1490  0.0101  0.0087  0.0349  0.0411  0.0435  0.0452  
\(\mathcal Q _\mathrm{B }^\mathfrak{d }\)  0.1490  0.0098  0.0038  0.0301  0.0311  0.0341  0.0356  
CH\(_4\)  \(Q^\mathrm{IMB }_\mathrm{C }\)  0.1603  0.5646  0.5957  0.5644  0.5689  0.5662  0.5660 
\(\mathcal Q _\mathrm{C }^\mathfrak{g }\)  0.1604  0.5908  0.6000  0.5695  0.5679  0.5666  0.5660  
\(\mathcal Q _\mathrm{C }^\mathfrak{d }\)  0.1604  0.5862  0.5946  0.5644  0.5644  0.5631  0.5625  
NH\(_3\)  \(Q^\mathrm{IMB }_\mathrm{N }\)  0.3174  0.7444  0.7541  0.7545  0.7829  0.7878  0.7883 
\(\mathcal Q _\mathrm{N }^\mathfrak{g }\)  0.3174  0.7471  0.7590  0.7597  0.7824  0.7882  0.7883  
\(\mathcal Q _\mathrm{N }^\mathfrak{d }\)  0.3174  0.7431  0.7535  0.7545  0.7748  0.7808  0.7809  
H\(_2\)O  \(Q^\mathrm{IMB }_\mathrm{O }\)  0.2604  0.7102  0.7178  0.7326  0.7596  0.7659  0.7663 
\(\mathcal Q _\mathrm{O }^\mathfrak{g }\)  0.2604  0.7123  0.7215  0.7359  0.7592  0.7662  0.7663  
\(\mathcal Q _\mathrm{O }^\mathfrak{d }\)  0.2604  0.7099  0.7177  0.7326  0.7524  0.7595  0.7596  
HF  \(Q^\mathrm{IMB }_\mathrm{F }\)  0.1553  0.4585  0.4738  0.4841  0.4972  0.4995  0.5000 
\(\mathcal Q _\mathrm{F }^\mathfrak{g }\)  0.1553  0.4600  0.4753  0.4851  0.4971  0.4996  0.5000  
\(\mathcal Q _\mathrm{F }^\mathfrak{d }\)  0.1553  0.4596  0.4738  0.4841  0.4932  0.4960  0.4963  
Averaged  \(\bar{\Delta }^\mathrm{IMB }_\mathrm{X }\)  0.3447  0.0379  0.0168  0.0156  0.0025  0.0003  0.0000 
Over all  \(\bar{\Delta }^\mathfrak{g }_\mathrm{X }\)  0.3447  0.0305  0.0140  0.0127  0.0029  0.0001  0.0000 
Molecules  \(\bar{\Delta }^\mathfrak{d }_\mathrm{X }\)  0.3447  0.0325  0.0170  0.0156  0.0074  0.0046  0.0043 
As follows from Table 4, using approximated freeatom orbitals in construction of IMBorbitals has no significant influence on calculated IMBatomic charges, especially for calculations involving the minimal set of basis functions. Indeed, the first column of numbers in Table 4 clearly indicates that within very poor basis set \(\mathfrak a \) atomic charges are particularly insensitive to quality of FAOs used in MOCprocedure (it becomes obvious if one recalls the results from Table 3).
On average, atomic charges \(\mathcal Q _\mathrm{X }^\mathfrak{d }\) and \(\mathcal Q _\mathrm{X }^\mathfrak{g }\) deviate from the corresponding exact charges \(Q^\mathrm{IMB }_\mathrm{X }\) by \(0.0029e\) and \(0.0020e\), respectively and thus, in the view of populationanalysis accuracy, they give rise to the same conclusions about the electronic structure of molecules. However, a more discerning comparison of atomic charges \(Q^\mathrm{IMB }_\mathrm{X }\) and \(\mathcal Q _\mathrm{X }^\mathfrak{g }\) reveals that the latter ones are usually closer to the corresponding exact IMBcharges (calculated within set of functions \(\mathfrak g \)). On the other hand, atomic charges \(\mathcal Q _\mathrm{X }^\mathfrak{d }\) deviate significantly from the exact charges only in the case of more accurate calculations (if we exclude BH\(_3\) molecule and take into consideration only basis set of TZVtype the average difference between the most exact values of \(\mathcal Q _\mathrm{X }^\mathfrak{d }\) and \(Q^\mathrm{IMB }_{\ mathrm{X}}\) assumes \(0.0043e\), e.i. about \(0.6\,\%\)). Indeed, convergence profiles that emerge from values of \(\bar{\Delta }_\mathrm{X }\) allow one to draw the conclusion that the more accurate are canonical FAOs used to construct a set of IMBorbitals the closer to the exact values (in a sense of basisset completeness) are the resulting IMBcharges calculated within any arbitrary basis set.
4 Summary
In this work we have introduced and briefly examined a simple method of generating a minimal set of moleculeadopted atomic orbitals. Contrary to the previously proposed method involving a reference set of (external) minimalbasis orbitals (e.g. Huzinaga’s MINI basis set) [7], in this approach we have used the criterion of maximum overlap to the set of freeatom orbitals obtained from ab initio calculations (using the same set of basis functions) for the corresponding system promolecule. Hence, the resulting minimalbasis orbitals are intrinsic for individual molecules and consequently exhibit appropriate convergence properties as the number of basis functions used in calculations increases.
It has also been demonstrated that the MOCscheme can be successfully utilized to approximate canonical freeatom orbitals within any arbitrary basis set using oneoff calculated FAOs of high quality (by default stored on disk); therefore, generating of IMBorbitals does not require everytime calculations of promolecular systems. Moreover, partial atomic charges calculated within representation of such moleculeadapted atomic orbitals tend to converge noticeably faster to their exact values in the limit of complete basis set.
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