Ab initio conical intersections for the Si(¹D) + H₂ reaction system: a lowest five singlet states study

Abstract

Conical intersection (CI) play a key role in the non-adiabatic processes and the geometric phase (GP) effect associated with CI have a significant influence on the quantum reactive scattering and bound-state calculations. In this work, the CIs and the GP effects associated with the CIs of the Si(¹D) + H₂ system are investigated using the internally contracted multireference configuration intersection method for the first time. The seams of the CIs (at the linear H–Si–H geometries, linear Si–H–H geometries and C_2v geometries) among the lowest five singlet states relative to the Si(¹D) + H₂ reaction system are searched systematically. The properties of minimum energy crossing points (MECPs) of these intersection seams are determined, the topographies of the potential energy surfaces in the branching spaces of MECPs and the GP effects associated with the CIs are discussed.

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1 Introduction

Silylene, SiH₂, the prototype radical in silylene family, plays an important role in the organic silicon chemistry,^1–3 and it is postulated as a reactive intermediate in the manufacturing process of amorphous silicon films by chemical vapor deposition.^4–7 The studies of SiH₂ started as early as 1967, when Dubois and co-workers observed electronic absorption spectra belonging to the X̃¹A₁–ùB₁ transition of this radical for the first time.⁸ It was not ascertained that whether the lower ¹A₁ state of the observed transition was the ground state or there was a lower-lying triplet state, as in CH₂, would be the ground state until Milligan and Wirsam's work confirmed that the lower state of the previously observed electronic transition of singlet SiH₂, is the ground state^9,10 and Apeloig and co-workers explained why CH₂ is a ground state triplet while SiH₂ is a ground state singlet.¹¹ Most of the studies about this radical concentrated on the X̃¹A₁ and ùB₁ states. Dubois and co-workers measured the absorption spectrum of the X̃¹A₁–ùB₁ transition of SiH₂ and pointed out the existence of a lot of strong and erratic rotational perturbations.^8,12,13 The anomalous rotational perturbations were interpreted as the combined effects of the Renner–Teller coupling between the two singlet states and the spin–orbit coupling with the lowest triplet state,^13,14 and the large amplitude motion of the bending vibration ν₂.^14,15 It was reported that fluorescence lifetime of SiH₂ in the ùB₁ state widely vary for each rovibronic level,^16,17 and this was interpreted in terms of mixing with adjacent levels of the ã³B₁ state,¹⁸ and the predissociation to Si(³P) + H₂.^16,19,20 Ishikawa et al. observed the stimulated emission pumping spectroscopy of the ùB₁–X̃¹A₁ transition and obtained information about the ã³B₁ state through the spin–orbit interaction.²¹ Yurchenko et al. first made an ab initio calculation of the potential energy surfaces (PESs) for the X̃¹A₁ and ùB₁ states, and they calculated the rovibronic energies of the two states using this ab initio PESs.²² Researches that refer to higher electronic states than the ùB₁ state were much fewer. The B̃¹A₁ state of SiH₂ and SiD₂ radicals was observed for the first time in 2005.²³ Before long, Tokue et al. evaluated the transition probabilities of the X̃¹A₁–ùB₁ and ùB₁–B̃¹A₁ systems of SiH₂ and SiD₂,²⁴ and they also investigated the photodissociation process of SiH₂(ùB₁) → SiH₂(B̃¹A₁) → Si(¹D) + H₂.²⁵

Conical intersections (CIs),²⁶ not isolated geometries but form a seam,²⁷ are widespread in polyatomic systems and play a key role in non-adiabatic processes,^28,29 such as charge-transfer reaction,^30,31 nonradiative transition, and electronic quenching. At a CI, the Born–Oppenheimer approximation stipulating that electronic and nuclear motions are separable breaks down.³² Therefore, molecular systems which exhibit such topologies rapidly switch between electronic states through the funnel of the CI.³³ Recently, many theoretical and experimental studies about the typical non-adiabatic process associated with CIs, such as electron transfer, isomerization, photoinduced unimolecular decay, and radiationless relaxation of the excited electronic state, have provided lots of valuable new insights into the nature of CIs.^30,34–41 There have been some ab initio studies about the CIs in triatomic molecules.^42–52 While, the detailed studies of the CIs or PES intersections of the Si(¹D) + H₂ system have never been reported.

Geometric phases (GP) or Berry phases⁵³ are ubiquitous in physics and chemistry, and some of the most fascinating phenomena in condensed matter science such as topological insulators,⁵⁴ ferroelectrics,⁵⁵ the Aharonov–Bohm effect,⁵⁶ and it is regarded as the signature property and associated non-adiabatic effect of CIs in molecules.^57,58 GP may arise when the Hamiltonian of a system depends on a set of parameters R. Consider a physical system described by a Hamiltonian that depends on time through a set of parameters, denoted by R = (R₁, R₂, …), i.e.,

H = H(R), R = R(t). (1)

We are interested in the adiabatic evolution of the system as R(t) moves slowly along a path C in the parameter space. For this purpose, it is useful to introduce an instantaneous orthonormal basis from the eigenstates of H(R) at each value of the parameter R, i.e.,

H(R)|n(R)〉 = ε_n(R)|n(R)〉. (2)

However, eqn (2) alone does not completely determine the basis function |n(R)〉, it still allows an arbitrary R-dependent phase factor of |n(R)〉. One can make a phase choice, also known as a gauge, to remove this arbitrariness. Here we require that the phase of the basis function is smooth and single valued along the path C in the parameter space.

According to the quantum adiabatic theorem, the solution of the time-dependent Schrödinger equation is written as

where the second exponential is known as the dynamical phase factor. Inserting eqn (3) into the time-dependent Schrödinger equationand multiplying it from the left by |ψ_n(t)〉, one finds that γ_n(t) can be expressed as a path integral in the parameter spacewhere _n(R) is a vector-valued function

This vector _n(R) is called the Berry connection or the Berry vector potential. Eqn (5) shows that, in addition to the dynamical phase, the quantum state will acquire an additional phase γ_n during the adiabatic evolution. Therefore, for a closed path, γ_n becomes a gauge-invariant physical quantity, known as the Berry phase or geometric phase in general; it is given by

By virtue of the above derivation of GP,⁵⁵ the GP associated with a CI comes down to the statement that the real adiabatic electronic wave function may acquire a phase which leads to a flip of sign when nuclear configuration traverse a closed path encircling a CI.^59,60 Since the total molecular wave function has to remain single-valued, a compensating sign change must occur in the nuclear motion wave function. In order to investigate the GP effects in chemical reaction dynamics and the energy levels of bound-state molecules, the compensating sign of the nuclear motion wave function must be included in quantum reactive scattering and bound-state calculations. The GP effects on the energy levels of bound-state molecules have been well understood,^60–62 it has been shown that they are pronounced on the bound-state spectrum of HO₂, Na₃, and N₃.^62–64 In recent years, some important theoretical and experimental studies make it more and more apparent that the GP effects can be significant in molecular scattering.^65–75 In chemical reactions at thermal energies the GP effects were vanishingly small over a wide range of energies due to the fact that they wash out when a summation or average over all contributing partial waves. While, the situation changes dramatically in an energy regime where only a single partial wave contributes to the reactions, this is the case for ultra-cold chemical reactions in which manifest GP effects occur and it can be modulated by applying an external electric field allowing the possibility of quantum control of chemical reactions in the ultra-cold regime.^73–75 In 2014, S. K. Min et al. demonstrated that there exist cases where a nontrivial GP present in BO approximation does not have a topological counterpart in the exact electron-nuclear problem. But they emphasized that it is not known whether these cases are the majority or the exception. Since, in practice, we normally cannot solve the full electron-nuclear problem, it will be extremely important, but difficult, to find simple mathematical criteria by which one can know if a nonvanishing BO-Berry phase survives in the exact electron-nuclear treatment.⁷⁶

The purpose of this work is to investigate the CIs and the associated GP effects among the five lowest singlet states of Si(¹D) + H₂ systems. The organization of the present article is as following. The details of the calculations are given in Sec. 2. The results are presented and discussed in Sec. 3. Finally, a brief summary is given in Sec. 4.

2 Theoretical approaches

2.1 Ab initio electronic structure calculations

The ab initio calculations presented in this work are performed with the MOLPRO suite of ab initio programs.⁷⁷ As the first step, the intersections are searched using the coupled-perturbed multi configuration self-consistent field (SA-CPMCSCF) method. For each calculation of locating the MECP between two electronic states, three vectors were evaluated at SA-CPMCSCF level: (1) non-adiabatic derivative coupling. (2) Gradient of the lower state. (3) Gradient of the upper state. Then, the internally contracted multireference configuration interaction (icMRCI) method was used for more sophisticated calculations of the single point energy. The icMRCI is a direct MRCI procedure performed in an internally contracted configuration basis, and the MRCI wavefunctions include all single and double excitations relative to complete active space self-consistent field (CASSCF) reference functions.^78,79 In this work the icMRCI is employed to describe the most correlation energy of single and double electron excitations, and further correlation energy due to higher excitations is approximated by the Davidson correction (+Q).⁸⁰ The SA-CASSCF is a MCSCF method that the high level multireference SCF energies and wavefunctions for a series of ground and excited states is calculate by a state-averaged variational calculation. In essence, the SA approach included each of the states in question with equal weight and the configuration space is equivalent for all states.^81,82 In this work, the state-averaged complete active space self-consistent field (SA-CASSCF) calculation with 0.2, 0.2, 0.2, 0.2, 0.2 weighting on the five states is performed to obtain the orbitals for the icMRCI and SA-CPMCSCF calculations. In the SA-CASSCF calculation, the average energy of the five states, which asymptotically correlate with the five degenerate Si(¹D) states, is optimized. The active space consists of six electrons distributed among six orbitals (6e, 6o), which corresponds to all valence electrons and valence orbitals. All the calculations are performed in C_s symmetry, three lowest state of A′ symmetry and two lowest state of A′′ symmetry are used for generating the internally contracted pairs in the icMRCI procedure. The basis set used is Dunning's correlation consistent polarized valence quadruple-zeta set augmented with diffuse functions^83–85 (aug-cc-pVQZ).

2.2 Characterization of the conical intersections

The topography of PES in the vicinity of a CI and the GP effect associated with the CI are discussed in the branching space of the CI. The branching space or g–h plane^26,51,86,87 is spanned by the orthonormal vectors i⃑ = g⃑/||g⃑|| and j⃑ = h⃑/||h⃑||, where g⃑ is the gradient difference, h⃑ is the derivative coupling vectors. The two vectors are determined by the analytic gradient technique in conjunction with the coupled-perturbed multi configuration self-consistent field SA-CPMCSCF method. Nuclear configurations in the branching space are characterized in terms of a set of natural polar coordinates in this plane, as shown in Fig. 1, r⃑ is the radial coordinate (r⃑ = x·i⃑ + y·j⃑), and φ is the angular coordinate or polar angle between i⃑ and r⃑. The CI is located at origin. Each r (we chose r = 0.01 : 0.01 : 0.2 in this work), together with the angular coordinate changing form 0 to 4π by the step 0.04π, define a trajectory of the nuclear configuration in the branching space. These nuclear configurations in the branching space are transformed into that in the Z-matrix before the electronic structure calculations. The PESs in branching space and the coefficient of the principal electron configuration of the related intersection electronic state are obtained from electronic structure calculations. Then, the double cone topographies of the PESs in the vicinity of the CI are evinced in the branching space, and the coefficient of the principal electron configuration of the related intersection electronic state along the motions of the nuclear configurations in the closed loop path around the CI in the two-dimensional specific branching space give rise to the GP effect.

Fig. 1 Schematic illustration of the branching space of a CI. i⃑ = g⃑/||g⃑||, j⃑ = h⃑/||h⃑||.

3 Results and discussion

3.1 Energy correlation diagram

As an analog of methylene, SiH₂ radicals have the similar energy correlation diagram for the lowest singlet states 1¹A′, 2¹A′, 3¹A′, 1¹A′′ and 2¹A′′, which correlate with the Si(¹D) + H₂ asymptote. The energy correlation diagram for the five singlet states is presented in Fig. 2. We can see that the 1¹A′ and 1¹A′′ states are bounded relative to the Si + H₂ dissociation limit, the depth of them are 60.86 kcal mol⁻¹ and 16.81 kcal mol⁻¹, respectively. While the 2¹A′, 3¹A′ and 2¹A′′ are above this limit. The resolutions of states of linear molecules into those of molecule in the C_2v or C_s symmetry, as shown in Table 1, indicate that the ¹A′ and ¹A′′ states will become the degenerate Δ_g or Π_g pairs at linearity geometries. Our calculations demonstrate that the 1¹A′, 1¹A′′ states and the 2¹A′, 2¹A′′ states become the degenerate Δ_g and Π_g pairs, respectively, at linearity, and hence there are Renner–Teller coupling between them. The 1¹A′ and 1¹A′′ states degenerate in the product asymptote and they are correlated with SiH(X²Π) + H products, the 2¹A′ and 2¹A′′ states degenerate in the product asymptote and they are correlated with SiH(X²Δ) + H products. The reaction Si(¹D) + H₂ → SiH(X²Π) + H have a significant endothermicity of 18.81 kcal mol⁻¹, which is very different from the most researched insertion-type reactions X + H₂ → HX + H (X = C, N, O, S) with more or less exothermicity.^88,89

Fig. 2 Energy correlation diagram for the Si(¹D) + H₂ reactive system. It is determined by MRCI + Q method, the energies are relative to Si(¹D) + H₂ asymptote.

Table 1 Resolution of states of linear molecules into those of molecule in the C_2v or C_s symmetry

Linear molecule^a	C2v (z → y)	Cs (σv → σ)
a The z axis in the linear case is assumed to lie in the internuclear direction.
Σ^+g	A1	A′
Πg	A2 + B2	A′ + A′′
Δg	A1 + B1	A′ + A′′

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The geometries of minima for the five singlet states of SiH₂ radical are summarized in Table 2 and compared with the available experimental and theoretical values.^8,12,15,23 The equilibrium SiH distances and the equilibrium HSiH angles of SiH₂ 1¹A′, 1¹A′′, 2¹A′ and 3¹A′ states agree well with the available experimental or theoretical values. To our knowledge, it is the first time that the geometry and the relative energy of SiH₂ 2¹A′′ are reported in detail.

Table 2 Equilibrium geometries and their relative energies of the lowest five singlet states of SiH₂. The energies are relative to the Si(¹D) + H₂ asymptote. Bond length in bohr, bond angle in degrees, and energies in kcal mol⁻¹

	Methods	RSiH	∠HSiH	Energies
a This work.b From ref. 8.c From ref. 12.d From ref. 15.e From ref. 23.
SiH2(1^1A′)	MRCI + Q^a	2.871	92.2	−60.86
	MRCI^b	2.870	92.2
	expt^c	2.874	92.1
	expt^d	2.865	92.1
	expt^e	2.861	92.1
SiH2(1^1A′′)	MRCI + Q^a	2.817	122.8	−16.81
	MRCI^b	2.816	122.7
	expt^c	2.806	122.0
	expt^d	2.810	123.0
SiH2(2^1A′)	MRCI + Q^a	2.777	165.1	16.34
	MRCI^b	2.776	164.8
SiH2(3^1A′)	MRCI + Q^a	3.160	180.0	36.37
	MRCI^b	3.115	180.0
SiH2(2^1A′′)	MRCI + Q^a	2.582	64.4	41.80

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3.2 Conical intersections

The CIs relative to the five lowest singlet states of Si(¹D) + H₂ asymptote are investigated systematically. The intersections covers CIs between 1¹A′ and 2¹A′ states, CIs between 2¹A′ and 3¹A′ states, at linear H–Si–H geometries; CIs between 1¹A′ and 2¹A′ states, CIs between 2¹A′ and 3¹A′ states, at linear Si–H–H geometries; CIs between 1¹A′′ and 2¹A′′ states, CIs between 2¹A′ and 3¹A′ states at C_2v geometries. The intersection seams of CIs are searched. Minimum energy crossing point (MECP) is an important point of an intersection seam. The energies and geometries of the MECPs on each intersection seams are reported in Table 3. The topographies of the PESs in the branching space of the MECPs and the GP effects associated with the MECPs are discussed. All these intersections are classified in terms of the symmetries of SiH₂ geometries and will be discussed in detail below.

Table 3 The geometries and relative energies of the minimum energy crossing points on the intersection seams among different states of the Si(¹D) + H₂ reactive system. The energies are relative to the Si(¹D) + H₂ asymptote, bond length in bohr, bond angle HSiH in degree and energy in kcal mol⁻¹, R_SiH2 > R_SiH1 at linear Si–H–H geometries

Classification	Electronic states	Geometries of MECPs	Energies of MECPs
H–Si–H	1^1A′(1^1A′′)–2^1A′(2^1A′′)	RSiH1 = 3.056, RSiH2 = 3.485	33.37
H–Si–H	2^1A′(2^1A′′)–3^1A′(2^1A′′)	RSiH1 = 3.160, RSiH2 = 3.160	36.37
Si–H–H	1^1A′(1^1A′′)–2^1A′(2^1A′′)	RSiH1 = 3.578, RHH = 1.952	24.06
Si–H–H	2^1A′(2^1A′′)–3^1A′(2^1A′′)	RSiH1 = 4.210, RHH = 1.370	11.13
C2v	1^1A′′–2^1A′′	RSiH = 3.165, ∠HSiH = 68.45	21.60
C2v	2^1A′–3^1A′	RSiH = 2.855, ∠HSiH = 60.68	118.13

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3.2.1 Conical intersections at linear H–Si–H geometries. The intersection seam between the 1¹A′ and 2¹A′ states and the energies along the seam are shown in Fig. S1(a) of the ESI.† The MECP on this seam locates at the geometry with R_SiH1 = 3.056 bohrs and R_SiH2 = 3.485 bohrs, as shown in Table 3, and the energy of this MECP is 33.37 kcal mol⁻¹. This MECP actually determines the special barrier hindering the H–Si–H bond fission to yield the SiH product and induce the barrier for H + SiH collinear hydrogen exchange reaction. The topographies of the PESs in the branching space of this MECP are shown in Fig. 3(a), it clearly indicates that this MECP is actually related to four (1¹A′, 2¹A′, 1¹A′′, 2¹A′′) states, so as the seam it locates on. When it comes to the linear geometry, the ¹A′, 1¹A′′ states and the 2¹A′, 2¹A′′ states become degenerate pairs respectively. The four states are degenerate at the MECP geometry and forming the double cone. The two lower cones (1¹A′ and 1¹A′′ states) surfaces in this region are in lack of the bound-states and the two upper cones (2¹A′ and 2¹A′′) surfaces form a local minima in this region. It should notice that there is a CI which belongs to the intersection seam between 2¹A′ and 3¹A′ on the upper solid surface which is corresponding to the 2¹A′ state, and with the nuclear configuration passing through this point in the negative direction of g⃑, the 2¹A′ and 2¹A′′ become non-degenerate states.

Fig. 3 The topography of potential energy surfaces in the branching space of the minimum energy crossing points of the intersections at linear H–Si–H geometries (a and b), linear Si–H–H geometries (c and d) and C_2v geometries (e and f). The corresponding nuclear configurations are distributed in the branching space with r = 0 : 0.01 : 0.2 and φ = 0 : 0.04π : 2π. The energies are relative to Si(¹D) + H₂ asymptote.

The intersection seam between the 2¹A′ and 3¹A′ states and the energies along the seam are shown in Fig. S1(b) of the ESI.† The MECP on this seam locates at the geometry with R_SiH1 = 3.160 bohrs R_SiH2 = 3.160 bohrs, which happen to be a C_2v geometry, as shown in Table 3, and the energy of this MECP is 36.37 kcal mol⁻¹. The topographies of the branching space PESs in the vicinity of this MECP are show in Fig. 3(b), it shows that this MECP is actually related to three (2¹A′, 3¹A′, 2¹A′′) states, so as the seam it located on. With that the nuclear configuration moving in the negative direction of g⃑, the components of the Π_g pair are 2¹A′ and 2¹A′′ states before the nuclear configuration reaching the MECP, and switch into 3¹A′ and 2¹A′′ states after that. The upper cone of this branching space PESs is also bounded in this region, while the lower cone is lack of bound-states in the vicinity of the MECP. It is obvious that there is a CI which belongs to the intersection seam between the 1¹A′ and 2¹A′ states on the lower cone, and the PES in the vicinity of this CI is bounded.

It has been mentioned that there is a CI point which belongs to the intersection seam between the 2¹A′ and 3¹A′ states on the branching space PES in the vicinity of the MECP between 1¹A′ and 2¹A′ states, conversely there is a CI point which belongs to the intersection seam between the 1¹A′ and 2¹A′ states on the branching space PES in the vicinity of the MECP between 2¹A′ and 3¹A′ states. In order to clarify the GP effects of these CIs, we extract the linear H–Si–H geometries out of the geometries of SiH₂ that distributing in the branching spaces. The intersection seams and the lines consisting of linear H–Si–H geometries extracted from the corresponding MECP branching spaces are shown in Fig. 4. The MECPs are the centers of the extracted linear geometries lines and other linear geometries with different r (r = 0 : 0.01 : 0.2) are symmetrically distributed with the MECPs. It shows that the cross point between the line that consists of the linear geometries extracted from the branching space of MECP between 1¹A′ and 2¹A′ states and the intersection seam that between the 2¹A′ and 3¹A′ states is located in the region with 0.09 < r < 0.10 and φ = π in the branching space of the MECP between 1¹A′ and 2¹A′ states. The cross point between the line that consists of linear geometries extracted from the branching space of the MECP between 2¹A′ and 3¹A′ states and the intersection seam that between 1¹A′ and 2¹A′ states is located in the region with 0.10 < r < 0.11 and φ = π in the branching space of the MECP between 2¹A′ and 3¹A′ states.

Fig. 4 The intersection seams at linear H–Si–H geometries and the line of linear H–Si–H geometries in the branching space (r = 0 : 0.01 : 0.2, φ = 0 : 0.04π : 2π) of MECP of corresponding seams.

It had been mentioned that with the nuclear configurations completing a closed path encircling a CI odd number of times, the sign of the corresponding real adiabatic electronic wavefunctions will change, which called GP effect. In this work the GP effects are discussed in the branching spaces of corresponding MECPs. Fig. 5 shows the coefficients of principal configuration of the related electronic states change along the closed paths consisting of two loops encircling the MECPs with different r (r = 0.01 : 0.01 : 0.2), and it present the GP effects in the title system. Three dimension representation of the GP effects are shown in Fig. S2 of the ESI.†

Fig. 5 Coefficient of the principal electron configuration of the related intersection electronic state along the paths correspond to r = 0.01 : 0.01 : 0.2 and φ = 0 : 0.04π : 4π in the branching space of minimum energy crossing points of intersection in Si(¹D) + H₂ system.

As for the 1¹A′ electronic state corresponding to the lower cone of the branching space PES of 1¹A′ and 2¹A′ states, all of the nuclear configuration paths with different r encircle only one CI, that is the MECP between 1¹A′ and 2¹A′ states, so the signs of the electronic wavefunctions corresponding to the paths with different r will change with the nuclear configurations completing the first closed loop path and change back to their original states with the second one, C(φ) = −C(φ + 2π), and this is shown in Fig. 5(a).

The 2¹A′ electronic state corresponding to the upper cone of the branching space PES of 1¹A′ and 2¹A′ involves two CIs, one is the MECP between 1¹A′ and 2¹A′ states and another is a common CI belonging to the intersection seam between 2¹A′ and 3¹A′ states and it is located in the region with 0.09 < r < 0.10 in this branching space. If r ≤ 0.09, the closed nuclear configuration paths encircle only one CI, and the signs of the corresponding electronic wavefunctions will change with the nuclear configurations completing the closed loop path and change back to their original states with the second one, C(φ) = −C(φ + 2π). While, if r ≥ 0.10, the closed paths encircle two CI points, and the signs of the corresponding electronic wavefunctions will not change with the nuclear configurations completing any number of loops, C(φ) = C(φ + 2π), and this is shown in Fig. 5(b).

The 2¹A′ electronic state corresponding to the lower cone of the branching space PES of 2¹A′ and 3¹A′ states, involves two CIs, one is the MECP between 2¹A′ and 3¹A′ states and another is a common CI belonging to the intersection seam between 1¹A′ and 2¹A′ states and it is located in the region with 0.10 < r < 0.11 in this branching space. If r ≤ 0.10, the closed nuclear configuration paths encircle only one CI, and the signs of the electronic wavefunctions corresponding to the paths with different r will change with the nuclear configurations completing the first loop and change back to their original state after the second one, C(φ) = −C(φ + 2π). While, if r ≥ 0.11, the closed paths encircle two CIs, and the signs of the corresponding electronic wavefunctions will not change with the nuclear configurations completing any number of loops. This is shown in Fig. 5(c).

As for the 3¹A′ electronic state corresponding to the upper cone of the branching space PES of 2¹A′ and 3¹A′ states, all of the nuclear configuration paths with different r encircle only one CI that is the MECP between 2¹A′ and 3¹A′, so the signs of the electronic wavefunctions corresponding to the paths with different r will change with the nuclear configurations completing the first loop and change back to their original states with the second one, C(φ) = −C(φ + 2π). This is shown in Fig. 5(d).

3.2.2 Conical intersections at linear Si–H–H geometries. The intersection seam between the 1¹A′ and 2¹A′ states at the linear Si–H–H geometries and the energies along the seam are shown in Fig. S1(c) of the ESI.† The MECP on this seam is located at the geometry with R_SiH1 = 3.578 bohrs R_HH = 1.952 bohrs, as shown in Table 3, and the energy of this MECP is 24.06 kcal mol⁻¹. This MECP actually determines the special barrier for the Si(¹D) collinearly approaching to the H₂. The topography of the branching space PES in the vicinity of this MECP is shown in Fig. 3(c), like the intersection at linear H–Si–H geometries, this MECP is actually related to four (1¹A′, 2¹A′, 1¹A′′, 2¹A′′) states, so as the seam it located on. The ¹A′, 1¹A′′ states and the 2¹A′, 2¹A′′ states become degenerate pairs, respectively, at linear geometries. Since that even at non-linear geometries the 1¹A′, 1¹A′′ states and the 2¹A′, 2¹A′′ states are very close to each other in energy, it seem that they are keep being degenerate pairs in all the MECP branching space region with r < 0.2 which is the largest r defined in this work. The four states are degenerate at the MECP geometry and forming the double cone. It is necessary to mention that there is a CI which belongs to the intersection seam between 2¹A′ and 3¹A′ states on the edge of the upper cone, as the relevant electronic states in this region are very close with each other in energies, it is not as obvious as that at linear H–Si–H geometries, but it will be clarified in the below paragraphs.

The intersection seam between the 2¹A′ and 3¹A′ states and the energies along the seam are shown in Fig. S1(d) of the ESI.† The MECP on this seam is located at the geometry with R_SiH1 = 4.210 bohrs and R_HH = 1.370 bohrs, as shown in Table 3, and the energy of this MECP is 11.13 kcal mol⁻¹. The topography of the branching space PES in the vicinity of this MECP is shown in Fig. 3(d), it should be similar with the intersection between the 2¹A′ and 3¹A′ states at the linear H–Si–H geometries that this MECP involve three (2¹A′, 3¹A′, 2¹A′′) states, so as the seam it located on. With the nuclear configuration moving in the negative direction of g⃑, the components of the Π_g pair are 2¹A′ and 2¹A′′ before the nuclear configuration reaching the MECP, and switch into 3¹A′ and 2¹A′′ after that, but this is less obvious compared with that at the linear H–Si–H geometries since that the three relevant states in this region are very close to each other in energy.

The intersection seams and the lines consisting of linear Si–H–H geometries extracted from the MECP branching spaces are shown in Fig. 6. The MECPs are the centers of the extracted linear geometries lines and other linear geometries with different r (r = 0 : 0.01 : 0.2) are symmetrically distributed with the MECPs. It shows that the cross point between the line that consists of linear geometries extracted from the branching space of the MECP between 1¹A′ and 2¹A′ states and the intersection seam that between 2¹A′ and 3¹A′ states is located in the region with 0.17 < r < 0.18 and φ = π in this branching space. There is not a cross point between the line that consists of linear geometries extracted from the branching space of the MECP between 2¹A′ and 3¹A′ states and the intersection seam that between 1¹A′ and 2¹A′ states.

Fig. 6 The intersection seams at linear Si–H–H geometries and the line of linear Si–H–H geometries in the branching space (r = 0 : 0.01 : 0.2, φ = 0 : 0.04π : 2π) of MECP of the corresponding seams.

As for the 1¹A′ electronic state corresponding to the lower cone of the branching space PES of 1¹A′ and 2¹A′ states, all of the nuclear configuration paths with different r encircle only one CI that is the MECP between 1¹A′ and 2¹A′ states, so the signs of the electronic wavefunctions corresponding to the nuclear configuration paths with different r will change with the nuclear configurations completing the first closed loop path and change back to their original states with the second one, C(φ) = −C(φ + 2π), and this is shown in Fig. 5(e).

The 2¹A′ electronic state corresponding to the upper cone of the branching space PES of 1¹A′ and 2¹A′ states involves two CIs, one is the MECP between ¹A′ and 2¹A′ states and another is a common CI belonging to the intersection seam between 2¹A′ and 3¹A′ states located in the region with 0.17 < r < 0.18 and φ = π in this branching space. If r ≤ 0.17, the closed loop paths of nuclear configurations encircle only one CI, so the signs of the electronic wavefunctions corresponding to the nuclear configurations paths with different r will change with the nuclear configurations completing the first closed loop and change back to their original states with the second one, C(φ) = −C(φ + 2π). While, if r ≥ 0.18, the closed loop paths encircle two CIs, and the corresponding electronic wavefunctions will not change sign with the nuclear configurations completing any number of loops, C(φ) = C(φ + 2π). That is shown in Fig. 5(f). All of this confirm that there is a CI on edge the upper cone of branching space PES of 1¹A′ and 2¹A′ states, and the CI is one point of intersection seam between 2¹A′ and 3¹A′ states.

As for both of the 2¹A′ and 3¹A′ electronic states corresponding to the lower cone and the upper cone of the branching space PES of 2¹A′ and 3¹A′ states, respectively, all of the nuclear configuration paths with different r encircle only one CI that is the MECP between 2¹A′ and 3¹A′ states. So, the signs of the electronic wavefunctions corresponding to the nuclear configuration paths with different r will change with the nuclear configurations completing the first closed loop and change back to their original states after the second one, C(φ) = −C(φ + 2π), and this is shown in Fig. 5(g) and (h).

3.2.3 Conical intersections at C_2v geometries. Two intersection seams are found at C_2v geometries of the title system. The intersection seam between the 1¹A′′ and 2¹A′′ states and the energies along the seam are shown in Fig. S1(e) of the ESI.† The MECP on this seam is located at the geometry with R_SiH = 3.165 bohrs and ∠HSiH = 68.45 degree, as shown in Table 3, and the energy of this MECP is 21.60 kcal mol⁻¹. The intersection seam between the 2¹A′ and 3¹A′ states is shown in Fig. S1(f) of the ESI.† The MECP on this seam locates at the geometry with R_SiH = 2.855 bohrs and ∠HSiH = 60.68 degree, as seen from Table 3, and the energy of this MECP is 118.13 kcal mol⁻¹. The topographies of the branching space PESs in the vicinity of the MECP between 1¹A′′ and 2¹A′′ states and that in the vicinity of the MECP between the 2¹A′ and 3¹A′ states are shown in Fig. 3(e) and (f). Both of the branching space PESs have the double cone structure. It seems that there are not any other CIs being involved in the two branching space PESs within r ≤ 0.2.

The GP effects associated with the MECP between 1¹A′′ and 2¹A′′ states are shown in Fig. 5(i) and (j). As for both of the 1¹A′′ state and the 2¹A′′ state, if r ≤ 0.07, the signs of the corresponding electronic wavefunctions will change with the nuclear configuration completing the first loop and change back to their original states with the second one, C(φ) = −C(φ + 2π). While, if r ≥ 0.08, the corresponding electronic wavefunctions will not change sign with the nuclear configurations completing any number of loops, C(φ) = C(φ + 2π). It indicate that there is a CI on lower and upper cones of the branching space PESs in the vicinity of MECP between 1¹A′′ and 2¹A′′ states in the region with 0.07 < r < 0.08, respectively. While this is not apparent in the topography of the branching space PES. We reason that there are some other electronic states that have not been taken into consideration in this work intersecting with the 1¹A′′, 2¹A′′ states.

The GP effects associated with the MECP between 2¹A′ and 3¹A′ states are shown in Fig. 5(k) and (l). As for both of the 2¹A′ electronic and 3¹A′ state, the signs of the corresponding electronic wavefunctions will change with the nuclear configurations completing the first closed loop and change back to its original states after the second one, C(φ) = −C(φ + 2π). It demonstrate that all of the nuclear configuration paths with different r encircle only one CI that is the MECP of intersection seam between 2¹A′ and 3¹A′ states. It is true that except the MECP there is not any other CI in the vicinity of the MECP within r ≤ 0.2 in the branching space.

4 Conclusions

CIs among the 1¹A′, 2¹A′, 3¹A′, 1¹A′′, 2¹A′′ states of the title system are investigated using the CASSCF or MRCI method. Finally several CI seams at the linear H–Si–H, linear Si–H–H and C_2v geometries are determined. The CI seams at the linear H–Si–H, Si–H–H geometries are accompanied by Renner–Teller coupling. The CI seams between 1¹A′ and 2¹A′ states actually involve four states: the 1¹A′, 1¹A′′ Renner–Teller coupling pair and the 2¹A′, 2¹A′′ Renner–Teller coupling pair. The CI seams between 2¹A′ and 3¹A′ states simultaneously involve 2¹A′′ state, for which the components of the Renner–Teller coupling pair are 2¹A′ and 2¹A′′ before the MECP, and switch into 3¹A′ and 2¹A′′ after the MECP.

The geometries and energies of the MECPs on these CI seams are reported, in particular, the MECP of CI seam between 1¹A′ and 2¹A′ states at the linear H–Si–H geometries is in the linear H–SiH dissociation direction and it induce the barrier for H + SiH collinear hydrogen exchange reaction with a height of 22.45 kcal mol⁻¹. The MECP of CI seam between 1¹A′ and 2¹A′ states at the linear Si–H–H geometries is located on the minimum energy reaction path of the Si(¹D) + H₂ collinear abstraction reaction, and it determines the energy barrier with a height of 24.06 kcal mol⁻¹.

The topographies of PESs in the vicinities of the MECPs are represented in their branching spaces. The double cone features of the branching space PESs around the MECPs at the linear H–Si–H geometries and the C_2v SiH₂ geometries are obvious. While, the double cone features of the branching space PESs around the MECPs at the linear Si–H–H geometries are subtle, as the related electronic states are close to each other in energy. In spite of that, the GP effects associated with the MECPs corroborate that the MECPs at the linear Si–H–H geometries are actually CIs, as with that at the linear H–Si–H and C_2v SiH₂ geometries.

The GP effects of these MECPs are demonstrated. The GP effects associated with the MECPs of intersection between 1¹A′′, 2¹A′′ states at the C_2v geometries indicate that there are some other electronic states that have not been taken into consideration in this work intersecting with the 1¹A′′, 2¹A′′ states.

We hope that this work will be helpful to providing intuitive understanding of the CIs and GP effects, and will contribute to the researches about diabatic dynamics with CIs and GP effects in the title system.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 21473218), the Chinese Ministry of Science and Technology (No. 2013CB834601), and the Beijing National Laboratory for Molecular Sciences (No. 20140160, 20150158) and Chinese Academy of Sciences.

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Footnote

† Electronic supplementary information (ESI) available: Seam lines of the conical intersections and three dimension representation of the geometric phase effect associated with the conical intersection in the Si(¹D) + H₂ system. See DOI: 10.1039/c7ra01021d

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