The Carbon, Silicon and Germanium Hydrides, XHn - ACS Publications

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Spin-Coupled Generalized Valence Bond Description of Group 14 Species: The Carbon, Silicon and Germanium Hydrides, XHn (n = 1− 4) Lu T. Xu, Jasper V. K. Thompson, and Thom H. Dunning, Jr.* Department of Chemistry, University of Washington, Seattle, Washington 98195

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S Supporting Information *

ABSTRACT: Although elements in the same group in the Periodic Table tend to behave similarly, the differences in the simplest Group 14 hydridesXHn (X = C, Si, Ge; n = 1−4)are as pronounced as their similarities. Spin-coupled generalized valence bond (SCGVB) as well as coupled cluster [CCSD(T)] calculations are reported for all of the molecules in the CHn/SiHn/ GeHn series to gain insights into the factors underlying these differences. It is found that the relative weakness of the recoupled pair bonds of SiH and GeH gives rise to the observed differences in the ground state multiplicities, molecular structures, and bond energies of SiHn and GeHn. A number of factors that influence the strength of the recoupled pair bonds in CH, SiH, and GeH were examined. Two factors were identified as potential contributors to the decrease in the strengths of these bonds from CH to SiH and GeH: (i) a decrease in the overlap between the orbitals involved in the bond and (ii) an increase in Pauli repulsion between the electrons in the two lobe orbitals centered on the X atoms. Finally, an analysis of the hybridization of the SCGVB orbitals in XH4 indicates that they are closer to sp hybrids than sp3 hybrids, which implies that the underlying cause of the tetrahedral structure of the XH4 molecules is not a direct result of the hybridization of the X atom orbitals.

1. INTRODUCTION Although there are many similarities between carbon, silicon, and germanium compounds, there are also significant differences. These differences are evident even in the simplest Group 14 moleculesthe carbon, silicon, and germanium hydrides, XHn (X = C, Si, Ge; n = 1−4). For example, the bonds in the lowest-lying excited states of (SiH, GeH) are markedly weaker than that in CH,1−3 the CH2 and (SiH2, GeH2) molecules have different ground state multiplicities and geometries,4,5 and the CH3 and (SiH3, GeH3) molecules have different ground state geometries.6 The differences are even more pronounced for other compounds of carbon, silicon and germanium; e.g., the ground state of C2H2 is linear but the ground states of Si2H2 and Ge2H2 have a “butterfly” structure7−9 and the ground states of Si2H4 and Ge2H4 have a trans-bent structure, not a planar geometry as in C2H4.10−15 Rationales have been offered for some of these differences based on molecular orbital and density functional theory,16−18 but the effect of electron correlation on the geometries and energetics of these species cannot be ignored.11 It has long been known that the chemistry of the first row elements (Li−Ne) differ from that of elements in subsequent rowsthe f irst row anomaly.19 In 1984 Kutzelnigg20 presented a detailed analysis of this anomaly, arguing that the increase in spatial separation of the Xns and Xnp orbitals between the first and following rows of the Periodic Table has a major impact on the hybridization of the X atom bond orbitals. He concluded that the resulting decrease in ns participation in the X atom bond orbitals was the root cause of the changes in © XXXX American Chemical Society

the chemical behavior of the main group elements down a column. We shall show that a preliminary analysis of the SCGVB orbitals of the (CHn, SiHn, GeHn) series finds that the X atom bond orbitals do, indeed, have more np character in (SiHn, GeHn) than in CHn. However, the situation is more complicated than put forward by Kutzelnigg. We shall show that differences in the bond energies in the carbon, silicon, and germanium hydrides also play an important role in the anomalies found in these molecules. Calculation of the equilibrium geometries and bond energies of the CHn, SiHn, and GeHn species using high-level electronic structure methods have been the subject of many studies over the past few decadesfar too many to describe here. However, few studies examined the trends found in the XHn series (X = C, Si, Ge) for n = 1−4, the exceptions including studies reported in refs 21 and 22 for the CHn series, refs 15 and 21−24 for SiHn series, and refs 15 and 25−28 for the GeHn series. None of these studies examined all three species together and few focused on the similarities and dissimilarities among the XHn species, although, in one study, Apeloig et al.29 analyzed the contributions to the energies of CH2 and SiH2 and concluded that the difference in the ground state multiplicity was a result of a complicated combination of factors involving the “frontier” electrons, the “core” electrons, and the nuclear repulsion energy. We will show that there is a Received: January 13, 2019 Revised: February 15, 2019

A

DOI: 10.1021/acs.jpca.9b00376 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

the singly occupied valence (active) orbitals. The active valence orbitals are, in general, nonorthogonal, although they are orthogonal to the doubly occupied core and valence orbitals. The spin function associated with the orbital product in eq 1 is a product of (αβ) spin functions for the electrons in a the doubly occupied orbitals and ΘnS,M for the electrons in the S a is a linear na singly occupied valence orbitals. The ΘnS,M S combination of all of the linearly independent ways, f nSa, to couple the spins of the na electrons to yield a state of total spin S and spin projection MS:41

simple explanation for this and other differences in the XHn series. In this paper we examine the variations in the electronic structure of the simplest Group 14 moleculesthe CHn, SiHn, and GeHn series (n = 1−4)using spin-coupled generalized valence bond (SCGVB) theory30−34 as well as accurate CCSD(T)/RCCSD(T) methods.35−37 The SCGVB calculations provide insights into the nature of the bonding in the XHn series, while the CCSD(T)/RCCSD(T) calculations provide a consistent, nearly quantitative description of the ground states and, for XH and XH2, the lowest-lying excited states of the XH n series. The CCSD(T)/RCCSD(T) calculations also serve to calibrate and/or validate the trends found in the SCGVB calculations and create a level of confidence in the insights offered by SCGVB theory. The molecular structures and bond energies of the ground and lowlying excited states of the XHn series were determined by successively adding hydrogen atoms to a lower hydride, i.e., XHn−1 + H → XHn. At each step the addition pathways were compared to gain insights into the similarities and dissimilarities in the resulting XHn species. Subsequent papers will examine other Group 14 compounds. The paper is organized as follows. In section 2 we briefly describe the theoretical and computational approaches used in this study. In section 3 we present and discuss (i) the basic SCGVB description of the XHn species, (ii) the results of the SCGVB and RCCSD(T)/CCSD(T) calculations on the XH series, and (iii) an analysis of the computational results for these species, comparing and contrasting the three species. Having shown that the trends observed in the RCCSD(T) calculations are reflected in the SCGVB calculation, we present an analysis of the SCGVB wave functions for the a4Σ− states of the CH, SiH, and GeH molecules in section 4 to gain insights into the underlying reason(s) for the dramatic weakening of the recoupled pair bonds in SiH and GeH, which underpins the dramatic differences in the structures and energetics of the XHn series. Finally, our conclusions are summarized in section 5.

ΘnSa, MS

v

a

S

(2)

i αβ − βα yzij αβ − βα yz zzjj zz··· α ··· ΘnSa, MS ; f na = jjj S 2 {k 2 { k

(3)

This spin function, which couples the spins of the electrons in the successive orbitals into singlet pairs and then high-spin couples those of the remaining electrons in the singly occupied orbitals, if any, represents the traditional Lewis structure of a molecule with the orbitals for the successive singlet-coupled pairs representing bond pairs or lone pairs. Although this spin coupling is dominant in most molecules around their equilibrium geometry, other spin couplings are important in isolated cases like C242 as well as, more commonly, in aromatic molecules.43 The SCGVB wave function is a product of (2nc + 2nv + na) orbitals and, as such, provides a well-defined orbital theory of the electronic structure of molecules. The SCGVB wave function has the advantage that it is more accurate than the Hartree−Fock (molecular orbital) wave function,31 accounting for nondynamical correlation ef fects in both atoms and molecules.44 As a result, the SCGVB wave function provides a more complete and consistent description of the electronic structure of atoms and molecules and is an excellent zero-order description of a broad range of molecules and molecular phenomena that are poorly described by a HF (MO) wave function, e.g., making and breaking chemical bonds, diradicals, and chemical reactions. For detailed expositions of SCGVB theory and its applications, the reader is referred to the papers reviewing SCGVB theory30−34 and papers on the fundamental aspects of covalent and recoupled pair bonding.45−49 The SCGVB wave function is equivalent to the f ull GVB wave function30,31 and to the spin-coupled valence bond wave function,32−34 both of which are now collectively referred to as Spin-Coupled Generalized Valence Bond (SCGVB) wave functions.50 One advantage of using the generally contracted correlation consistent basis sets is that the first s and p orbitals in the basis set correspond to the atomic HF orbitals. Thus, if the X atom SCGVB valence orbitals are dominated by their valence ns and np components, n = 2 (carbon), 3 (silicon), 4 (germanium), which they usually are, then the coefficients of these functions, c(ns) and c(np), can be used as a rough measure of the

̂ ϕ ...ϕ ϕ ϕ ϕ ...ϕ ϕ φ φ ...φ αβ ...αβαβ ...αβ ΘnSa, M = (ϕ c1 c1 cn cn v1 v1 vn vn a1 a2 an S v

∑ cS;k ΘnS ,M ;k

Kotani spin functions are used in all of the SCGVB calculations reported here. The Kotani spin functions are orthonormal; so, na the squares of the coefficients in the ΘS,M expansion S correspond to the fraction (or weight, wk) of the associated spin functions in the total SCGVB wave function: wk = cS;k2. Of particular interest is the weight of the perfect pairing spin function:

ΨSCGVB c

=

k=1

2. THEORETICAL AND COMPUTATIONAL METHODS In this paper we combine two approaches that we have found useful in our past studies:31 (i) calculations with the coupled cluster singles and doubles method with a perturbative treatment of triples [CCSD(T), RCCSD(T)],35−37 to provide a nearly quantitative description of the species and states of interest, and (ii) calculations with spin-coupled generalized valence bond (SCGVB) theory30−34 to provide insights into the underlying cause(s) of any differences in the electronic structure of the species of interest. To ensure accurate solutions of the CCSD(T)/RCCSD(T) and SCGVB equations, we use augmented correlation consistent basis sets of quadruple quality (aug-cc-pVQZ) for C and H38 as well as Ge,39 and the corresponding tight d-function augmented set [aug-cc-pV(Q+d)Z] for Si.40 The SCGVB wave function for an atom or molecule has the general form:

c

f Sna

a

(1)

In eq 1 the {ϕci} refer to the doubly occupied core orbitals, the {ϕvi} to the doubly occupied valence orbitals, and the {φai} to B

DOI: 10.1021/acs.jpca.9b00376 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A hybridization of these orbitals. In this paper we take 100 × c(ns)2/[c(ns)2 + c(np)2] as a measure of the percentage of ns character in the SCGVB orbital and 100 × c(np)2/[c(ns)2 + c(np)2] as a measure of the percentage of its np character. Of course, the SCGVB orbitals are changed from their atomic forms in a number of ways, including contraction/expansion, polarization, and delocalization. These changes introduce an element of ambiguity into these percentages and must be kept in mind in interpreting the computed results. For the SCGVB calculations the bond energies were computed by subtracting the energy of the XHn molecule at Re from that of XHn−1 + H at large separation (R = 100 Å). This recognizes the fact that SCGVB calculations on different states of XHn may not yield exactly the same energies at very large R and that these energies may differ slightly from the sum of the energies of the two atoms. This well-known effect results from the constraints on the SCGVB wave function imposed by the different symmetries of the states involved and is usually addressed by referencing the energies to XHn−1 + H at large separation. The CCSD(T)/RCCSD(T) wave function describes an important subset of nondynamical correlation effects as well as dynamical correlation effects, both of which are critical for quantitative predictions of molecular properties. This assumes that the zero-order wave function is well described by a Hartree−Fock wave function, although it has been found that modest deviations from this requirement, e.g., as exhibited in the 1A1 states of XH2, have only minor effects on the predicted results.51 In combination with the quadruple-ζ correlation consistent basis sets, the CCSD(T)/RCCSD(T) wave function typically yields geometries accurate to a couple of milliÅngstroms and energy differences accurate to a couple of kcal/ mol. The accuracy provided by the CCSD(T)/RCCSD(T) method is of special importance for the species of interest here that lack reliable experimental data. All of the calculations presented in this study were performed with the Molpro suite of quantum chemical programs (version 2010.1).52,53 The CASVB module in Molpro was used to perform the SCGVB calculations,54,55 while the coupled cluster module was used to perform the CCSD(T)/RCCSD(T) calculations. None of the calculations reported herein include relativistic effects. Although this omission will lead to small, but non-negligible errors in the calculated quantities and these errors will be larger for SiHn and GeHn than for CHn, the nonrelativistic calculations provide a consistent description of the Group 14 hydrides of interest and will reveal any important differences in the electronic structure of the XHn series.

atomic orbitals and spin coupling coefficients resulting from molecular formation. Thus, atomic orbitals play a fundamental role in SCGVB theory. The HF valence electronic configurations for the ground states of the carbon, silicon, and germanium atoms are ns2npx1npy1. However, in all of these atoms a second configuration involving an unoccupied npz orbital makes an important contribution to the atomic wave function. This is a direct result of the sp near-degeneracy effect.56,57 The wave function that takes this effect into account for the Group 14 atoms (omitting the core orbitals) is: ΨMCSCF[X(3 P)] = c Xns (̂ ϕXnsϕXnsϕXn p ϕXn p αβαa x

y

− c Xn p (̂ ϕX′ n p ϕX′ n p ϕXn p ϕXn p αβαa z

z

x

y

(4)

In eq 4 we explicitly noted that the radial function for the Xnpz orbital correlating the Xns orbital pair, ϕ′Xnpz, differs from that for the singly occupied atomic Xnpx and Xnpy orbitals. We find the two types of orbitals to be similar with the correlating Xnpz′ orbitals being more concentrated in the valence region than the atomic Xnp orbitals (see Figure S1 in the Supporting Information). For all three atoms, the impact of the second configuration in eq 4 is significant with the energy lowering being larger for C (11.98 kcal/mol) than for Si (8.19 kcal/ mol) and Ge (7.52 kcal/mol); see Table 1. As noted by Table 1. Results of SCGVB Calculations on the Ground, 3P, States of the Carbon, Silicon, and Germanium Atoms: Energy Lowerings, Coefficient Ratios, and Lobe Orbital Overlaps (SCGVB Energy Lowering in kcal/mol) ESCGVB − EHF

(% ns, % np)

S(Xns−, Xns+)

11.98 8.19 7.52

(86.8%, 13.2%) (87.7%, 12.3%) (88.5%, 11.5%)

0.737 0.754 0.770

C(3P) Si(3P) Ge(3P)

Clementi,57 taking the sp near-degeneracy effect into account results in a smooth, monotonic increase in the remaining correlation energy, i.e., the dynamical correlation energy, for the first-row atoms from Be to Ne. The two-configuration wave function in eq 4 is the natural orbital form of the (un-normalized) SCGVB wave function for these atoms (again, omitting the core orbitals): i αβ − βα zy zzαα ΨSCGVB[X(3 P)] = (φXns φXns φXn p φXn p jjj z− z+ x y 2 { k

(5)

where

3. RESULTS AND DISCUSSION In this section we summarize the results of the SCGVB and CCSD(T)/RCCSD(T) calculations on the ground states of the atoms X and the ground states of XHn with X = C, Si, and Ge for n = 1−4. For XH and XH2, results are also reported for the lowest-lying excited states of these species. Of particular interest are the differences in the spin multiplicities, molecular geometries, and bond energies of the ground states in the XHn series. Additional information on the SCGVB calculations on the carbon, silicon, and germanium atoms and CHn, SiHn, and GeHn molecules is provided in the Supporting Information. 3.1. Ground States, 3P, of the X Atoms. In SCGVB theory, it is found that, by and large, molecules are composed of perturbed atoms, with clearly delineated changes in the

φXns = z±

c Xn p

c Xns ϕ ± c Xns + c Xn p Xns

c Xns + c Xn p

= c XnsϕXns ± c Xn pϕX′ n p

ϕX′ n p

z

(6a)

z

and φXn p = ϕXn p x

x

φXn p = ϕXn p y

y

(6b)

For symmetry reasons, the perfect pairing spin function, the spin function in eq 5, has a weight of unity for all of the Group 14 atoms. Because of their shapes, the first two orbitals in the SCGVB wave function, (φXns−, φXns+), are referred to as ns lobe orbitals C

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The Journal of Physical Chemistry A and form the Xns± lone pair. Although the lobe orbitals are spn hybrids, they are very weak hybrids with cXnp/cXns = 0.390 (C), 0.375 (Si), and 0.360 (Ge), which corresponds to only ∼12% np character; see Table 1. The two lobe orbitals have a nonzero overlap given by: c Xns − c Xn p S(Xns−, Xns+) = c Xns + c Xn p (7) The overlaps vary from 0.737 (C) to 0.770 (Ge); see Table 1. The overlap is a measure of the spatial separation of the two lobe orbitals, with smaller overlaps indicating larger spatial separations. In Figure 1, the two lobe orbitals and the singly

Figure 2. Orbital diagrams representing the formation of XH(X2Π) and XH(a4Σ−) from X(3P) + H(2S). The black dots in the orbital represent the number of electrons in the orbital. The lines connecting the two Xns± lobe orbitals in the atom indicate that the spins of the electrons in these orbitals are singlet coupled.

formation of the ground states, X2Π, and first excited states, a4Σ−, of XH (X = C, Si, Ge) from the X(3P) + H(2S) separated atoms. The X2Π ground states are formed by singlet coupling the spins of the electrons in the Xnpz and H1s orbitals, forming a traditional (polar) covalent bond. The a4Σ− states are formed by recoupling the spins of the electrons in the (Xns−, Xns+) lone pair to form a new singlet-coupled bond pair (Xns+, H 1s), leaving the third electron in the “leftover” Xns− orbital. Thus, the bond in the a4Σ− state is a recoupled pair bond.48,49,58−60 The SCGVB wave function provides the quantitative foundation for the symbolic orbital diagram representation of the XH(X2Π) and XH(a4Σ−) states in Figure 2, and the valence SCGVB orbitals for the XH(X2Π) and XH(a4Σ−) states at their respective Re’s are plotted in Figure 3. Although the atomic orbitals are clearly affected by molecular formation, e.g., contracting, polarizing, delocalizing, and rotating, there is a clear connection between the XH SCGVB orbitals of the XH molecule in Figure 3 and the SCGVB orbitals of the X atoms in Figure 1 as well as to the SCGVB orbital diagrams in Figure 2. There are five spin functions that contribute to the X2Π state 5 and four that contribute to the a4Σ− state in in Θ1/2,1/2

Θ53/2,3/2 . However, at Re the SCGVB wave functions for both states are dominated by the perfect pairing spin functions: ̂ ′ φ′ φ′ φ′ φ′ ΨSCGVB(X2 Π) = (φ Xns Xns Xn p H1s Xn p

Figure 1. Contour plots of the (ns−, ns+) lobe orbitals and the npx and npy orbitals for the C, Si, and Ge atoms along with the corresponding orbital diagrams (without black dots that represent the electron occupancy). The orbital diagram for the npy orbital, which is perpendicular to the plane of the paper, is just a shaded circle. Contours are shown from 0.05 to 0.25 in increments of 0.05; red contours are positive, blue contours negative.

i αβ − βα yzij αβ − βα yz zzjj zzα × jjj 2 {k 2 { k −

+

z

y

̂ ′ φ ′ φ ′ φ ′ φ ′ ijjj αβ − βα yzzzααα ΨSCGVB(a4 Σ−) = (φ Xns+ H1s Xns− Xn px Xn py j 2 z{ k

(8a)

(8b)

occupied npx and npy orbitals are plotted for the three atoms. The silicon and germanium orbitals are more diffuse than the carbon orbitals, but their shapes are otherwise quite similar. As can be seen, the two Xns± lobe orbitals are largely localized on opposite sides of the atom, making it relatively easy to form a bond with a ligand even though the spins of the electrons in these orbitals are singlet coupled. Thus, the SCGVB description of the X atoms, with four singly occupied orbitals, foreshadows the tetravalence of the carbon, silicon, and germanium atoms. The SCGVB orbitals of the carbon, silicon, and germanium atoms are represented schematically by the SCGVB orbital diagrams, which are shown in the last row in Figure 1. 3.2. Ground, X2Π, and First Excited, a4Σ−, States of XH. The SCGVB orbital diagram in Figure 2 describes the

In eq 8, primes are used to indicate that the SCGVB orbitals, although identified by their atomic lineage, have been changed by molecular formation. Note also that in eq 8b the X atom has been rotated so that the Xns± lobe orbitals now lie on the zaxis as in Figures 2 and 3. The weights of the perfect pairing spin function (wPP) at Re are (0.993, 0.964) in CH, (0.995, 0.977) in SiH, and (0.995, 0.964) in GeH for the (X2Π, a4Σ−) states. Thus, the bonds in both states correspond, by and large, to traditional singlet-coupled electron pairs: a (Xnpz′, H1s′) bond pair in the X2Π state and a (Xns+′, H1s′) bond pair in the a4Σ− state. There is a second electron pair in the XH(X2Π) statethe lone pair on the X atom, (Xns−′, Xns+′). On the basis of the magnitude of wPP, the electron pairs in the X2Π state are more strongly singlet coupled than the bond pair in the recoupled pair bond. This is undoubtedly due to the D

DOI: 10.1021/acs.jpca.9b00376 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

In the XH(X2Π) states, the favorable overlaps of the lone pair orbitals (Xns−′, Xns+′) are less than in the atoms by just 0.02−0.04, while the bond pair overlaps (Xnpz′, H1s′) are nearly the same for all of the species, 0.80−0.82. In addition to these overlaps, there are unfavorable overlaps associated with the lone pair orbitals and those in the bond pair, i.e., the (Xns−′, Xnpz′) and (Xns−′, H1s′) overlaps. The magnitudes of these two overlaps are smaller for CH (0.26, 0.17) than for SiH (0.36, 0.20) and GeH (0.38, 0.24). Since these overlaps give rise to Pauli repulsion between the lone pair and the bond pair, as R decreases, the atomic Xns± lobe orbitals rotate away from the internuclear region in order to reduce the repulsion between the two pairs.30 At Re for each species, the angle between a line drawn through the maxima in the lone pair orbitals and a line through the X atom nucleus is 125−130° for all three diatomic molecules. In the XH(a4Σ−) states, the favorable overlaps of the SCGVB bond orbitals (Xns+′, H1s′) are slightly larger than in the XH(X2Π) states for CH (+0.02) but smaller for SiH (−0.03) and GeH (−0.03). There are unfavorable overlaps between the left-over Xns−′ orbitals and the orbitals in the bond pairs. The overlaps between the two SCGVB orbitals on the X atoms, (Xns−′, Xns+′) range from 0.51 to 0.60 for CH− GeH and are much larger than those for (Xns−′, H1s′), which range from 0.21 to 0.27 with the overlaps being largest for CH. Although the (Xns−′, Xns+′) overlap is much larger than the lone pair-bond pair overlaps in the X2Π state and undoubtedly impacts the strength of the bond in the a4Σ− state, it is smaller than the corresponding overlap in the atoms (see Table 1). This is as expected since the (Xns−′, Xns+′) overlap is favorable in the atoms and unfavorable in the molecules. We will more carefully examine the SCGVB wave functions for the XH(a4Σ−) states in section 4. As a direct result of molecular formation, the Xnpz′ bond orbital in the XH(X2Π) state builds in ns character (hybridizes) as R decreases in order to increase its amplitude in the internuclear region and, therefore, its overlap with the H1s′ orbital. Since this orbital at Re is dominated by its ns and np atomic orbital components, the percentage of atomic ns and np character in the SCGVB orbital can be estimated from c(ns) and c(np) as noted in the last section. At their respective Re’s, we find that the X atom bond orbital is 39% 2s and 61% 2p in CH, 46% 3s and 54% 3p in SiH, and 37% 4s and 63% 4p in GeH. Thus, the X atom bond orbital has substantial ns character in all three molecules with somewhat less ns character in CH and GeH than in SiH. The hybridization of

Figure 3. Contour plots of the SCGVB valence orbitals for the XH(X2Π) states and the XH(a4Σ−) states at their respective Re’s. The orbital name indicates its atomic origin; the prime signifies that the orbital has been modified by molecular formation. In the top figure the Xnpy′ orbitals are plotted in the yz plane. Contours are shown for 0.05 to 0.25 in increments of 0.05; red contours are positive, blue contours negative.

polarizing effect of the three electrons with high-spin coupling in the a4Σ− state. Except for symmetry reasons, the SCGVB active valence orbitals have nonzero overlaps. Some of the overlaps lower the energy and are favorable, e.g., those involving the orbitals in a bond or a lone pair, while other overlaps give rise to Pauli repulsion and are unfavorable (although repulsion may be a bit of a misnomer, it is commonly used). The latter includes overlaps of the orbitals involved in different bonds or between a lone pair and a bond. The overlaps of the SCGVB orbitals for all of the XHn molecules are included in the Supporting Information. In this article we will only summarize these results.

Table 2. Calculated and Experimental Bond Distances, Re, and Dissociation Energies, De(X−H), for the Ground, X2Π, and Lowest-Lying Excited States, a4Σ−, of CH, SiH, and GeHa Re (Å) CH(X Π) CH(a4Σ−) Δ(a-X) SiH(X2Π) SiH(a4Σ−) Δ(a-X) GeH(X2Π) GeH(a4Σ−) Δ(a-X) 2

De (kcal/mol)

SCGVB

CC

exptl

SCGVB

CC

1.131 1.091 −0.040 1.538 1.494 −0.044 1.617 1.560 −0.057

1.120 1.089 −0.031 1.523 1.496 −0.027 1.603 1.569 −0.034

1.1198 1.085 −0.035 1.5197

70.90 63.02 −7.88 64.23 34.62 −29.61 57.75 22.98 −34.77

83.38 66.55 −16.83 73.35 34.65 −38.70 67.59 25.63 −41.96

1.58724 1.5834 −0.0038

exptl 83.9 66.8 70.1−71.6

a

The CC heading refers to CCSD(T) or RCCSD(T) as appropriate. E

DOI: 10.1021/acs.jpca.9b00376 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A the npz′ bond orbital is clearly evident in the orbital plots in Figure 3, where most of the blue (negative) contours associated with the np atomic orbitals are missing. For CH, the lone pair orbitals are dominated, as expected, by 2s character, 78%. The lone pair orbitals in the SiH and GeH states have significant contributions from the other p functions in the basis set and, thus, the c(np) coefficient cannot be used in this analysis. However, c(ns) increases monotonically from CH to GeH, from 0.864 to 0.888 to 0.916, suggesting an increase in the ns contribution to the lone pair orbitals from CH to GeH. As expected from the orbital diagrams, the ns character of the Xns+′ SCGVB bond orbital is much larger in the a4Σ− state than in the X2Π state. For the XH(a4Σ−) state the percentage of ns character in the bond orbital is estimated to be 77% (CH), 74% (SiH), and 73% (GeH). The percentage of ns character in the left-over Xns−′ orbital increases from 78% in CH to 85% in SiH and 82% in GeH, the inverse of the change in the bond orbital. Calculated Geometries and Energetics of XH. Table 2 summarizes the calculated and experimental bond distances (Re) and bond energies (De) for the X2Π and a4Σ− states of the three XH species along with the differences in the bond distances and energies between the two states. For CH, the RCCSD(T) calculations predict a bond energy (De) of 83.38 kcal/mol for the X2Π state and 66.55 kcal/mol for the a4Σ− state with ΔDe = De(a4Σ−) − De(X2Π) = −16.83 kcal/mol. These predictions are in excellent agreement with the experimental values of 83.9 kcal/mol (X2Π) and 66.8 kcal/ mol (a4Σ−), with ΔDe = −17.1 kcal/mol.1 Thus, in CH the recoupled pair bond in the a4Σ− state is only slightly weaker than the covalent bond in the X2Π state. For the X2Π state the SCGVB calculations predict the bond energy to be 12.48 kcal/ mol less than that from the RCCSD(T) calculations, but only 3.53 kcal/mol less for the a4Σ− state, yielding ΔDe(SCGVB) = −7.88 kcal/mol. Because the spins of three of the five electrons in the valence orbitals in the a4Σ− state are high-spin coupled, the SCGVB wave function provides, as expected, a much better description of this state than the X2Π state. As an interesting aside, we note that RHF calculations with the same basis set predict the wrong energetic ordering of the two states, with the 2 Π state being 6.38 kcal/mol above the 4Σ− state. The SCGVB calculations on the CH(X2Π) state predict a bond length (Re) of 1.131 Å, which is 0.011 Å longer than that predicted by the RCCSD(T) calculations, 1.120 Å. The latter agrees well with experiment (1.1198 Å61). For the a4Σ− state the calculated and experimental bond lengths are 1.091 Å (SCGVB), 1.089 Å [RCCSD(T)], and 1.085 Å (exptl61). The RCCSD(T) calculations predict shorter bond distances than the SCGVB calculations for both states, although the inclusion of dynamical correlation decreases the bond length in the lowspin CH(X2Π) state (−0.011 Å) far more than in the high-spin CH(a4Σ−) state (−0.002 Å). This again illustrates the unique character of the a4Σ− state with three of the five electrons in the valence orbitals being high-spin coupled. The calculated differences in the bond lengths of the two states, ΔRe = Re(a4Σ−) − Re(X2Π), are −0.040 Å (SCGVB) and −0.031 Å [RCCSD(T)]. The bond energies for the SiH states predicted by the SCGVB calculations are 64.23 kcal/mol for the X2Π state and 34.62 kcal/mol for the a4Σ− state. The inclusion of dynamical correlation in the RCCSD(T) calculations leads to a significant increase in the bond energy for the X2Π state, by 9.12 to 73.35

kcal/mol, but has essentially no effect on the bond energy for the a4Σ− state, by 0.03 to 34.65 kcal/mol. The calculated differences in the a4Σ− and X2Π bond energies (ΔDe) are −29.61 kcal/mol (SCGVB) and −38.70 kcal/mol [RCCSD(T)]. In SiH, unlike CH, the recoupled pair bond in the a4Σ− state is far weaker than the covalent bond in the X2Π state. As a result, RHF calculations on SiH predict the correct ordering of the 2Π and4Σ− states in SiH with the a4Σ− state 19.74 kcal/ mol higher in energy than the X2Π state. The prediction of De(X2Π) = 73.35 kcal/mol from the RCCSD(T) calculations is larger than any of the values for the experimental dissociation energies quoted by Feller and Dixon,24 which range from 70.1 to 71.6 kcal/mol. However, this value is in good agreement with the value that they recommend, 73.4 ± 0.4 kcal/mol. For SiH, the SCGVB calculations predict the bond distance in the X2Π state to be 1.538 and 1.494 Å for the a4Σ− state (ΔRe = −0.044 Å). These bond length predictions are in reasonable agreement with those from the RCCSD(T) calculations, 1.523 and 1.496 Å (ΔRe = −0.027 Å), which predicts a bond length for the X2Π state in good agreement with that determined from experiment (1.5197 Å61). As expected, the magnitude of the RCCSD(T)−SCGVB difference is smaller for the a4Σ− state (+0.002 Å) than for the X2Π state (−0.015 Å). The effect of dynamical correlation on Re(a4Σ−) is predicted to be opposite to that for Re(X2Π), namely, an increase rather than a decrease. The bond dissociation energies for the GeH states predicted by the SCGVB calculations are 57.75 kcal/mol for the X2Π state and 22.98 kcal/mol for the a4Σ− state. Following the trend observed in CH and SiH, the RCCSD(T) calculations predict a significant increase in the bond energy for the X2Π state, by 9.84 to 67.59 kcal/mol, but only a minor increase for the a4Σ− state, by 2.65 to 25.63 kcal/mol. As in SiH, the recoupled pair bond in the GeH(a4Σ−) state is much weaker than the covalent bond in the GeH(X2Π) state: ΔDe = −34.77 kcal/mol (SCGVB) and −41.96 kcal/mol [RCCSD(T)]. There are few data on the bond energy of GeH(X2Π) from experiment. Huber and Herzberg62 quote a value of 81 kcal/ mol, but this is undoubtedly too high. The high-level relativistic calculations of Li et al.3 predict De(X2Π) = 66.2 kcal/mol, which agrees reasonably well with the prediction from the current nonrelativistic calculations. RHF calculations again predict the correct ordering of the 2Π and 4Σ− states with the energy gap being 26.58 kcal/mol. For GeH, the SCGVB calculations predict the bond distance in the X2Π state to be 1.617 Å and that in the a4Σ− state to be 1.560 Å. These bond lengths differ by +0.014 and −0.009 Å from those predicted by the RCCSD(T) calculations, 1.603 Å (X2Π) and 1.569 Å (a4Σ−). The experimental bond length for the X2Π state is shorter than that predicted by the RCCSD(T) calculations by a couple of milli-Å (1.58724 Å63), which is largely a result of core−valence correlation and relativistic effects. Li et al.3 report a value of 1.582 Å when both of these effects are included. The RCCSD(T) calculations predict the bond length in the a4Σ− state to be 0.034 Å shorter than in the X2Π state. Again, the inclusion of dynamical correlation shortens the bond length in the GeH(X2Π) state and lengthens that in the GeH(a4Σ−) statethe same trend as in SiH and the opposite to that in CH. The experimental bond distance for the a4Σ− state is 1.5834 Å.63 Analysis of Computational Results for XH. Although the X2Π and a4Σ− states arise from the same separated atom limit, F

DOI: 10.1021/acs.jpca.9b00376 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A the strength of the bond in the a4Σ− state is expected to be less than that in the X2Π state. This is a direct result of the recoupling of the spins of the electrons in the Xns± lobe orbitals when the bond in the a4Σ− state is formed, resulting in an electron in the “left-over” Xns−′ orbital that has a large overlap with the Xns+′ bond orbital. The difference in the two bond energies is a result of this trade-off as well as the inherent differences between forming a bond with an Xnp orbital versus an Xns+ orbital and other factors, e.g., Pauli repulsion between the electron(s) in the X lone pair orbital(s) and the bond pair and the changes in the exchange interactions.64 The RCCSD(T) calculations predict that the strengths of the covalent bonds (De) in the X2Π state decreases by 10.03 kcal/mol from CH to SiH and by another 5.76 kcal/mol from SiH to GeH. These relatively modest decreases reflect the expected weakening of the XH bond down the column in the Periodic Table, a result of the increased bond distances, the more diffuse bond orbitals on the X atoms, and the decrease in the exchange interaction as well as other factors. The recoupled pair bonds in the a4Σ− states of SiH and GeH, however, are far weaker than that in CH: 34.65 kcal/mol (SiH) and 25.63 kcal/mol (GeH) versus 66.55 kcal/mol (CH), i.e., ΔDe(XH−CH) = −31.90 kcal/mol (X = Si) and −40.92 kcal/ mol (X = Ge). These trends are mimicked by the SCGVB calculations where for the X2Π state ΔDe(XH−CH) = −6.67 kcal/mol (X = Si) and −13.15 kcal/mol (X = Ge) and for the a4Σ− state ΔDe(XH−CH) = −28.40 kcal/mol (X = Si) and −40.04 kcal/mol (X = Ge). As we shall see, the dramatic weakening of the recoupled pair bond in SiH and GeH has a marked effect on the ground state multiplicities, molecular geometries, and bond energies in the XHn (n = 2, 3) molecules. The SCGVB calculations predict the differences between the bond lengths (Re) in the X2Π and a4Σ− state to be −0.040 Å (CH), −0.044 Å (SiH), and −0.057 Å (GeH). The RCCSD(T) calculations predict these differences to be somewhat smaller: −0.031 Å (CH), −0.027 Å (SiH), and −0.034 Å (GeH). Because the SCGVB wave function provides a better description of the high-spin a4Σ− state than the lowspin X2Π state, the differences predicted by the SCGVB calculations are larger than those predicted by the RCCSD(T) calculations. Although the bond distances in the XH(a4Σ−) state are expected to be shorter than in the XH(X2Π) states because of the increased ns component of the X atom bond orbital in the a4Σ− state, the differences predicted by the SCGVB and RCCSD(T) methods differ substantially from the differences in the spatial extents of the ns and np orbitals from atomic RHF calculations,65 ⟨r(np)⟩ − ⟨r(ns)⟩ = 0.066 Å (C), 0.29 Å (Si), and 0.34 Å (Ge). Although the bond lengths clearly depend on the X atom orbital sizes, the dif ferences in the sizes of the atomic Xns and Xnp orbitals have only a small effect on the differences in the XH bond lengths. Although the De’s and Re’s from the SCGVB calculations differ from those predicted by the RCCSD(T) calculations, the trends are the same in both cases; see Figure 4. Of particular importance is the fact that the SCGVB calculations predict the dramatic weakening of the recoupled pair bond in SiH and GeH versus the corresponding bond in CH. In section 4 we will examine the SCGVB wave functions for the a4Σ− states of CH, SiH, and GeH more closely in an attempt to identify the underlying cause of the dramatic weakening of the recoupled pair bonds in SiH and GeH. 3.3. Ground and Lowest Excited (1A1, 3B1) States of XH2. Figure 5 illustrates the formation of the ground and first

Figure 4. Variation of De and Re for the XH series (X = C, Si, Ge) as predicted by the SCGVB (■) and RCCSD(T) (●) calculations for the X2Π (solid lines) and a4Σ− (dashed lines) states.

excited states of XH2 from XH(X2Π) + H(2S). Formation of a second XH covalent bond with the electron in the Xnpπ′ orbital of XH(X2Π) gives rise to the XH2(1A1) states, while formation of a recoupled pair bond with the electron in the Xns+′ lobe orbital of XH(X2Π) gives rise to the XH2(3B1) state. Although the two bonds in the XH2(3B1) state are different at R(HX−H) = ∞, resonance will cause the two XH bonds to become equivalent as R(HX−H) decreases to Re and the (Xns−′, Xns+′) → (Xns+′, H1s′) pair recoupling is complete. Since the 1A1 state results from the formation of a covalent bond and the 3B1 state from the formation of a recoupled pair bond, the ordering of these two states of XH2 will depend on the relative strengths of these two bonds, although resonance between the covalent bond and the recoupled pair bond will decrease the expected energy gap. As we found in section 3.2, the covalent and recoupled pair bonds in CH are of comparable strength, ΔDe = 16.83 kcal/mol, but in SiH and GeH the recoupled pair bonds are weaker than the covalent bonds by 38.70 kcal/mol in SiH and 41.96 kcal/mol in GeH. Thus, we expect the 1A1 and 3B1 states to be close in energy in CH2, but the 1A1 state is expected to be favored in SiH2 and GeH2.

Figure 5. Orbital diagrams representing the formation of the XH2(1A1, 3B1) states from XH(X2Π) + H(2S). In the 1A1 state the H atom approaches the XH molecule above the plane of the molecule. In the last frame of this figure, the atoms in the XH2 molecule have been rotated into the plane of the paper with the lone pair orbitals above and below the molecular plane. The lines connecting the two Xns± lobe orbitals in the XH molecule indicate that the spins of the electrons in these orbitals are singlet coupled. G

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between the two sets of bond orbitals are |S(Xb1′, Xb2′)| = (0.32, 0.26, 0.28), |S(Xb1′, H1s2′)| = (0.12, 0.16, 0.19), and | S(H1s1′, H1s2′)| = (0.05, 0.09, 0.11) for (CH, SiH2, GeH2) and the overlaps between the bond orbitals and lone pair orbitals are |S(Xlp+′, Xb1′)| = (0.20, 0.30, 0.34) and S(Xlp+′, H 1s1′) = (0.13, 0.17, 0.22). As can be seen, unlike the favorable overlaps, the unfavorable overlaps in the three species differ significantly for the 3B1 and 1A1 states with the overlaps between the two bond orbitals on the X atom, |S(Xb1′, Xb2′)|, being larger in the 3B1 state than in the 1A1 state. Finally, in the 3 B1 state the overlap between the X atom unpaired orbital and bond orbital, |S(Xns−′, Xb1′)|, is nearly as large as that between the two X atom bond orbitals. The valence SCGVB orbitals for the XH2(1A1) and XH2(3B1) states are plotted in Figure 6. As expected, the X

The orbital diagrams in Figure 5 can also be used to gain insight into the geometries of the 1A1 and 3B1 states of XH2. Since the 1A1 states arise from the formation of a second covalent bond involving the Xnpπ′ orbital of XH(X2Π), the nominal bond angles in the XH2(1A1) state are 90°. Pauli repulsion will increase the bond angle, more so in CH2 with its shorter bonds and greater electronegativity than in SiH2 and GeH2. We expect the nominal bond lengths in the XH2(1A1) states to be similar to those in the XH(X2Π) states. Since the 3 B1 states result from the formation of recoupled pair bonds involving the Xns+′ orbitals of XH(X2Π) and the Xns+′ orbitals are oriented at an angle of 125−130° with respect to the bond axis for the three species, the nominal bond angles of the XH2(3B1) states are 125−130°. However, at (Re, θe) the bonds are a resonance mixture of a recoupled pair bond and a covalent bond, so we expect θe < 125−130°. Again, the bond angle is expected to be larger in CH2(3B1) than in SiH2(3B1) and GeH2(3B1). The presence of a resonance-enhanced recoupled pair bond implies that the nominal bond length in the XH2(3B1) state will be approximately the average of the Re’s for XH(X2Π) and XH(a4Σ−). There are five spin functions that can contribute to the 1A1 state in Θ60,0 and nine that can contribute to the 3B1 state in Θ61,1. However, at Re the SCGVB wave functions for both states are dominated by the perfect pairing spin functions: ̂ ′ φ ′ φ ′ φ′ φ ′ φ′ ΨSCGVB(1 A1) = (φ Xlp Xlp Xb H1s Xb H1s

i αβ − βα yzij αβ − βα yzij αβ − βα yz zzjj zzjj zz × jjj 2 {k 2 {k 2 { k −

+

1

1

2

2

(9a)

̂ ′ φ′ φ ′ φ′ φ′ φ′ ΨSCGVB(3 B1) = (φ Xb H1s Xb H1s Xns Xn p i αβ − βα yzij αβ − βα yz zzjj zzαα × jjj 2 {k 2 { k 1

1

2

2

−

y

(9b)

The weights of the perfect pairing spin function at the equilibrium geometries of the (1A1, 3B1) states are (0.988, 0.950) in CH2, (0.996, 0.985) in SiH2, and (0.995, 0.982) in GeH2. Thus, the bonds in both of these states correspond to traditional singlet-coupled electron pairs. As in XH, the electron pairs in the low-spin state of XH2 are more strongly singlet coupled than in the high-spin state, although the differences are not as great as in the monohydrides, a result that is consistent with the smaller fraction of electrons that are high-spin coupled. The most important favorable overlaps in the 1A1 and 3B1 states of XH2 are those involving the orbitals in the bond pairs, which are 0.80−0.82 in the 1A1 states and 0.80−0.78 in the 3B1 states of XH2, varying by just ±0.02 with X for both states. The overlaps of the lone pairs in the 1A1 states, 0.67 for CH2/SiH2, are also nearly identical and are 0.06−0.08 less than in the atoms. The lone pairs in GeH2 have an overlap of 0.72. There are also a number of unfavorable orbital overlaps in the SCGVB wave functions of XH2. In the 3B1 states these correspond to (i) overlaps between the two sets of bond orbitals|S(Xb1′, Xb2′)| = (0.48, 0.42, 0.51), |S(Xb1′, H1s2′)| = (0.16, 0.18, 0.23), and |S(H1s1′, H1s2′)| = (0.08, 0.08, 0.11) for (CH, SiH2, GeH2)and (ii) overlaps between the bond orbitals and the unpaired Xns−′ orbital|S(Xns−′, Xb1′)| = (0.41, 0.51, 0.59) and |S(Xns−′, H1s1′)| = (0.21, 0.22, 0.27) for the same sequence. In the 1A1 states the unfavorable overlaps

Figure 6. Contour plots of the SCGVB valence orbitals for the XH2(1A1) states and the XH2(3B1) states at their respective (Re, θe). Only one of the two bond pairs is plotted (Xb1′, H1s1′). Contours are shown for 0.05−0.25 in increments of 0.05; red contours are positive, and blue contours are negative.

atom bond orbital has more np character in the XH2(1A1) state, (62%, 58%, 66%), than in the XH2(3B1) state, (30%, 33%, 32%), for (CH2, SiH2, GeH2). This is in line with the fact that both bonds in the 1A1 state involve the Xnp′ orbitals and the bonds in the 3B1 state are a resonance mixture of the Xnp′ orbital and the Xns+′ orbital. As expected, the lone pair orbitals in the XH2(1A1) states have more ns character than the bond orbitals. For SiH2 and GeH2, the percentages are 76% and 83%, respectively. There is significantly less ns character in the H

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Table 3. Calculated and Experimental Geometries, (Re, θe), and Dissociation Energies, De(HX−H), for the Ground and Lowest-Lying Excited States for CH2, SiH2, and GeH2a θe (deg)

Re (Å) CH2(X3B1) CH2(a1A1) Δ(a-X) SiH2(X1A1) SiH2(a3B1) Δ(a-X) GeH2(X1A1) GeH2(a3B1) Δ(a-X)

De (kcal/mol)

SCGVB

CC

exptl

SCGVB

CC

exptl

SCGVB

CC

exptl

1.088 1.122 0.034 1.535 1.494 −0.041 1.611 1.554 −0.057

1.078 1.109 0.031 1.517 1.480 −0.037 1.596 1.545 −0.051

1.075 1.107 0.032 1.514 1.485 −0.029 1.5934

131.0 102.1 −28.9 94.4 117.7 23.3 93.3 118.9 25.6

133.6 102.0 −31.6 92.3 118.3 26.0 91.6 119.4 27.8

133.8 102.4 −31.0 92.1 122.4 30.3 91.3

93.71 82.98 10.73 72.05 53.90 18.15 64.90 43.27 21.63

105.66 96.38 9.28 79.76 59.28 20.48 73.34 49.89 23.45

106.5 97.5 9.0 76−82

56.4

The De’s are relative to the XH2 ground state: X3B1 for CH2 and X1A1 for SiH2 and GeH2. The CC heading refers to CCSD(T) or RCCSD(T) as appropriate. a

L

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The favorable overlaps in the ground state, X1A1, of XH4 involve the orbitals in the bond pairs: (0.70, 0.81, 0.80) for (CH4, SiH4, GeH4) with the overlap in the CH4 molecule being decidedly less than that in SiH4/GeH4. The unfavorable orbital overlaps are the overlaps between the bond orbitals| S(Xb1′, Xb2′)| = (0.51, 0.30, 0.38), |S(Xb1′, H1s2′)| = (0.10, 0.14, 0.19), and |S(H1s1′, H1s2′)| = (0.14, 0.06, 0.10) for (CH4, SiH4, GeH4). As in XH3, the unfavorable overlaps again differ more significantly than the favorable overlaps with the overlaps of the X atom orbitals in CH4 being particularly large. There are two important observations: (i) the overlap of the bond orbitals, |S(Xb1′, H1s1′)|, are much less in CH4 than in (SiH4, GeH4), and (ii) the overlap of the X atom bond orbitals, |S(Xb1′, Xb2′)|, is much larger in CH4 than in (SiH4, GeH4). The same trend was found in XH3, but the differences are magnified in XH4. Two of the four bond orbital pairs are plotted in Figure 8. An analysis of the atomic components of the X atom bond

(GeH3), whereas the SCGVB calculations predict a difference of just 1.24 kcal/mol, 7.63 (SiH3), and 6.39 kcal/mol (GeH3). From an analysis of the infrared laser diode spectra, Yamada and Hirota79 deduced a barrier height of 5.34 kcal/mol for SiH3. From an analysis of the resonance enhanced multiphoton ionization spectra of GeH3, Johnson et al.82 estimated the barrier height in GeH3, including zero-point energy, to be 4.4 kcal/mol. Analysis of Computational Results for XH3. The most important trend in the XH3 series is the change in the nature of the new H2X−H bond, along with the resulting change in the geometry of the XH3 molecules. The H2C−H bond is an inplane resonance-enhanced covalent bond. As a result, the H2C−H bond is strong (116.37 kcal/mol) and the CH3 molecule is planar. The (H2Si−H, H2Ge−H) bonds, however, result from the formation of a recoupled pair bond involving the electrons in the Xns+′ lone pair orbitals, which point above the plane of the XH2, and the H 1s orbital. Although these bonds will be enhanced by resonance with the two covalent bonds in XH2, they are still weaker than the covalent bonds in the XH2(X1A1) states, by 5.72 kcal/mol in SiH3 and 9.14 kcal/ mol in GeH3. The formation of a recoupled pair bond in SiH3 and GeH3 leads to pyramidal geometries with predicted (nonrelativistic) bond angles of ∼111°. Our preliminary predictions based on the SCGVB orbital diagram was that the XH3(2A1) states would have bond lengths slightly less than that of XH2(X3B1) state for X = C and slightly less than that for the XH2(X1A1) state for X = Si and Ge. This prediction holds true for X = Si and Ge, where ΔRe = −0.037 Å (SiH3) and −0.052 Å (GeH3), but the distances are the same for the X3B1 state of CH2 and the X2A1 state of CH3, 1.078 Å. Both SiH3 and GeH3 have modest barriers to inversion, slightly over 5 kcal/mol for both species, very similar to that in ammonia (NH3), 5.06 kcal/mol,83 although NH3 has a pair of electrons in its lone pair orbital and SiH3/GeH3 have a single electron in that orbital. 3.5. Ground, X1A1, States of XH4. The ground states of XH4 result from the straightforward formation of a covalent bond with the remaining unpaired, singly occupied orbital of XH3(X2A1). Again, resonance will cause the four XH bonds to become equivalent as R(H3X−H) decreases to Re. To minimize Pauli repulsion between the bond pairs, the geometry of the resulting molecule is tetrahedral. We expect the equilibrium bond lengths in XH4 to be close to those in XH3. There are 14 spin function functions that contribute to the X1A1 state in Θ80,0. However, at (Re, θe) the SCGVB wave functions for all of the XH4 species are, once again, dominated by the perfect pairing spin function:

Figure 8. Contour plots of the SCGVB valence orbital for XH4(X1A1). Only one of the four bond pairs has been plotted. Contours are shown for 0.05−0.25 in increments of 0.05; red contours are positive, and blue contours are negative.

orbitals shows that they have nearly equal ns and np character in SiH4 and GeH4: (ns, np) = (54%, 46%) in SiH4 and (52%, 48%) in GeH4. The estimated amount of 2p character in the C atom bond orbital in CH4, 37%, is much less than in (SiH4, GeH4). The smaller amount of np character in the C atom bond orbital is consistent with the larger overlap between the orbitals noted above. It is also consistent with the preference in (SiH4, GeH4) to form bonds with their np orbitals. In standard valence bond treatments of these molecules the X atom bond orbitals would be considered to be sp3 hybrids derived from the X(nsnp,3 4S) state.84−86 However, the ratios found here, although rough estimates, are much closer to the values that would be expected from the C(ns2np2, 3P) state, which would predict a 50:50 ratio. In the SCGVB description of the X atoms, the four singly occupied orbitals in the X atom, eq 5, also correspond to a nearly equal occupancy of the ns and np orbitals. This result is consistent with the results reported by Cook,87 who analyzed the SCVB calculations of Penotti et al.88 and found that the carbon atom bond orbitals were ∼sp0.9 hybrids. Calculated Geometries and Energetics of XH4. Table 7 summarizes the calculated and experimental bond distances (Re) and bond dissociation energies (De) for the X1A1 states of

̂ ′ φ′ φ′ φ′ φ′ φ′ φ′ φ′ ΨSCGVB(1 A1) = (φ Xb H1s Xb H1s Xb H1s Xb H1s

i αβ − βα yzij αβ − βα yzij αβ − βα yzij αβ − βα yz zz zzjj zzjj zzjj × jjjj 2 z{jk 2 z{jk 2 z{jk 2 z{ k (11) 1

1

2

2

3

3

4

4

The weights of the perfect pairing (PP) spin function at the equilibrium geometry of the XH4(X1A1) states are 0.907 (CH4), 0.990 (SiH4), and 0.979 (GeH4). Thus, once again, the bonds in these states correspond to traditional singlet-coupled electron pairs, although the weight of the perfect pairing function is clearly less in CH4 than in (SiH4, GeH4); this is similar to the trends in XH2 and XH3. M

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Table 7. Calculated and Experimental Bond Distances, Re, and Dissociation Energies, De(H3X−H), for the Ground State for CH4, SiH4, and GeH4a Re (Å) CH4(X1A1) SiH4(X1A1) GeH4(X1A1)

De (kcal/mol)

SCGVB

CC

exptl

SCGVB

CC

exptl

1.102 1.499 1.557

1.088 1.479 1.541

1.0857 1.4734 1.5158

97.43 85.33 78.78

112.08 96.35 89.09

112.5 95.6