The BNB state considered to be of symmetry broken (SB) character has been studied by high level multireference variational and full configuration interaction methods. We discuss in great detail the roots of the so-called SB problem and we offer an in depth analysis of the unsuspected reasons behind the double minimum topology found in practically all previous theoretical investigations. We argue that the true reason of failure to recover a D∞h equilibrium geometry lies in the lack of the correct permutational symmetry of the wavefunctions employed and is by no means a real effect.
Received 01 March 2013Accepted 20 May 2013Published online 11 June 2013
The author would like to thank Professor B. T. Sutcliffe for his questions that sharpened the argument and his thoughtful interest in this work. Professor T. H. Dunning, Jr. is thanked for reading the manuscript. The author expresses his gratitude to Professor E. R. Davidson for critically reading and commenting on the manuscript. I am truly indebted to Professor M. D. Morse for his clarifications on the experimental part of this problem and for our discussions on symmetry related subjects. A three month stay at the University of Stuttgart where work on related problems was done was supported by a grant from the HPC – Europa 2 project (Project No. 211437) with the support of the European Community – Research Infrastructure Action of the FP7. The author would like to thank Professor H.-J. Werner and the entire Werner group for the interesting discussions we had on symmetry issues.
Article outline: I. INTRODUCTION II. THE BNB SAGA III. RESULTS AND DISCUSSION IV. CONCLUSIONS
1.The proper translation of the Greek work symmetria (from the prefix syn [together] and the noun metro [measure]) is commensurability or in proportion with. During the classical era, the Greeks interpreted this word as the harmony of the different parts of an object, the good proportions between its constituent's parts. In some way, symmetry was always related to beauty, truth, and good. In a generalized and contemporary meaning, one can speak about symmetry of an “object” if under any kind of transformation at least one property of the “object” is left invariant.
2.The germ of the idea of symmetry in natural philosophy appeared in Astronomia Nova, Kepler's 1609 book in which his first two laws of planetary motion are described. In 1619, he published Harmonices Mundi where he attempted to explain the geometrical forms and physical phenomena in terms of harmony and congruence.
3.In a celebrated passage of his Dialogo, Galilei enunciated what is now called the Galilean relativity principle that forms the foundation of Newtonian mechanics.
4.E. Noether, Nachr. D. König. Gessesch. D. Wiss. Zu Göttingen, math-phys. Klasse, 235 (1918)
4.E. Noether, N. D. König, G. D. Wiss, and Z. Göttingen, [Transp. Theory Stat. Phys.1, 186 (1971)].
4.The curious reader should consult the marvelous book Les Théorèmes de Noether: Invariance et lois de Conservation au XXe Siècle by Y. Kosmann-Schwarzbach and L. Meersseman, Éditions de l’ École Polytechnique, 2006, for its authoritative historical and scientific commentary on both of Noether's theorems.
5.In most of the cases, additional constraints are imposed on approximate wavefunctions due to their incomplete form. These constraints are often related to the point group of the associated geometry. It should be mentioned though that an exact solution will have the correct symmetry of the problem, e.g., in the Kepler problem all points of the phase space will be such that both the angular momentum and the so-called Laplace-Runge-Lenz vectors will be conserved without imposing them from the outset. In the atomic case, the Hartree-Fock orbitals are restricted to be eigenfunctions of l2, lz, s2, sz while orbitals with the same value of l2 are constrained to have the same radial part.
6.E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959), p. 259, formula 22.20b and the paragraph that follows it. The German edition of the book appeared in 1931.
7.As an example, consider the Li atom, its RHF representation is ∼|1s(1)a(1)1s(2)β(2)2s(3)a(3)|, the assumed spatial function 1s(1)1s(2)2s(3) can be coupled only to the spin eigenfunction and not to the second one , so RHF does not conform to the Wigner's way of writing down a properly antisymmetric wavefunction being simultaneously a spin eigenfunction. The spatial part of its UHF representation can be in general coupled to both spin eigenfunctions. By applying a spin projection operator to UHF, we finally get the wavefunction written the Wigner's way.
8.A wavefunction which is not invariant under all symmetry operations of the point group is said to be symmetry broken (SB). In classical mechanics, SB occurs when a stable minimum undergoes bifurcation and splits into two stable minima. Although the above cases are distinct, we will use the term in both of its meanings.
9.J. M. L. Martin, J. P. François, and R. Gijbels, J. Chem. Phys.90, 6469 (1989).
22.Both the and diabatic states of the LVC model were assumed to have the same harmonic frequency while the separation constant Δ was evaluated by considering the experimental energy gap T0 = 0.785 eV and the calculated zero point energies of both adiabatic states for a given value of the interaction parameter λ. But most importantly they fitted experimental data on a completely decoupled (1D instead of 3D) model that is parametrized by a three parameter 2 × 2 matrix. Taking into account the assumptions and oversimplifications made the barrier ΔE(D∞h ← C∞v) of 18 cm−1 should be viewed with extreme caution.
23.H. Ding, M. D. Morse, C. Apetrei, L. Chacaga, and J. P. Maier, J. Chem. Phys.125, 194315 (2006).
24.The bond length was obtained by measuring the rotational constant of the ground vibrational state, converting this to a moment of inertia, then assuming a rigid symmetric linear structure, and solving for the bond length. However, even in its ground vibrational level there is a ZPE motion, and for this molecule the bending motion will have fairly large amplitude motion because of its small bending frequency. This means that in the (0, 0, 0) vibrational level, the two B atoms spend a significant amount of time with a shorter distance between them than they do when they are in the linear configuration. This decreases the vibrationally averaged moment of inertia, which makes the bond length computed from the measured rotational constant shorter than what is found at the equilibrium position. Private communication with Professor M. D. Morse; see Ref. 23.
25.In Ref. 17, we read “The physical reason of this instability in terms of bonding is that the two electronic states, which are orthogonal in the high−symmetry configuration, become mixed under the odd nuclear displacements thus producing additional (covalent) energy gain that offsets the losses by distortion.” We believe that the argumentation presented in Sec. III C of Ref. 17 is of tautological nature. Both electronic states at all points of the configurational space are orthogonal and not only in the high symmetry nuclear configuration. Their highly correlated methods give rise to a double well curve, using this fact as a starting point in the 2 × 2 problem they gain nothing new. They get as an output what they entered as an input. The fundamental question is whether the double minimum well is correct or not and not to rename the configuration interaction as a pseudo Jahn−Teller effect.
26.It was mistakenly reported as .
27.As was also done in Ref. 12, the model employed decouples all modes of vibration and uses a three parameter 2 × 2 matrix for the description of the – system.
28.S. L. Guberman and W. A. GoddardIII, Chem. Phys. Lett.14, 460 (1972), discussions on a spatially projected wavefunction are older and given by several authors;
31.It is interesting to consider the following results on the N(4Su) atomic state based on different CASSCF/cc-pVQZ calculations. The RHF energy is ERHF = −54.400 176 Eh. When an orbital of s type is added into the active space, the CASSCF energy obtained is E(2s+2p+s) = −54.407 805 Eh. The same energy is obtained by a spin coupled wavefunction. The ∼7 mEh energy lowering is traditionally attributed to the radial correlation of the electron pair. The true origin though is the restoration of the permutational symmetry (A. G. H. Barbosa and M. A. C. Nascimento, Int. J. Quantum Chem.99, 317 (2004)).
31.The coefficients of the four linearly independent spin functions associated to a 5e−/2S + 1 = 4 problem are c1 = 0.9740 (the HF spin function), c2 = 0.1688, c3 = 0.1194, and c4 = 0.0924. Although the c2–c4 coefficients are extremely small their presence is vital for the permutational invariance of the total wavefunction; see Ref. 6. When orbitals of p symmetry are included in the active space, then the GVB-like optimized wavefunction has an energy E(2s + 2p + s + p) = −54.409 152 Eh, while the inclusion of d type orbitals has a dramatic effect in the energy stabilization, E(2s+2p+s+p+d) = −54.433 463 Eh. In all these wavefunctions, the additional three p and five d orbitals were used as the GVB complement of the ∼2sN one and not in a CASSCF way. For a “traditional” approach to the subject, see J. W. Hollett and P. M. W. Gill, J. Chem. Phys.134, 114111 (2011);
32.The geometry optimization of the first root of 2Σ+ symmetry of a two state SACASSCF([11e−/13orb = (2s + 2p)B × 2 + (2s + 2p + s′)N] wavefunction gives rise to a SB structure with = 1.3606 Å, = 1.3057 Å, and E = −103.908 589 Eh. Since it is hard to optimize the GVB companion of the ∼2sN orbital in the a1 symmetry block, we “forced” the additional orbital to be of the “correct shape” by including two states of 12A1 symmetry in the optimization procedure. This way it is certain that the electrons will be in different orbitals along the σ frame of the molecule. We tested the energy loss due to the above followed SACASSCF procedure by calculating the CASSCF (5e−/5orb) + 1 + 2 (+Q)/cc-pVQZ (C1 symmetry) energy of the N(4Su) atomic state based on orbitals optimized for a 4A reference [E = −54.521 480 (−54.525 51) Eh] and on orbitals optimized for a 6A reference [E = −54.519 195 (−54.528 06) Eh]. In the molecular BNB case, the energy loss is ∼4 mEh.
33.It is well known that the HF equations being nonlinear admit many solutions for the same nuclear configuration. It is indeed possible to find a symmetry adapted (SA) solution for the BNB state at the RHF level of theory; = −103.704 892 Eh with = = 1.3013 Å and positive harmonic frequencies ω(ss)/ω(as)/ω(b) = 1270/2178/9 cm−1, while for the symmetry broken (SB) structure = −103.742 274 Eh with = 1.2470 Å, = 1.3669 Å and positive harmonic frequencies ω(ss)/ω(as)/ω(b) = 1190/2060/178 cm−1. A geometry optimization at the CISD, CISD + Q, ACPF, and RCCSD(T) levels of theory based on either the SA or SB RHF solutions give a D∞h or a C∞v structure, respectively. A one state CASSCF wavefunction is practically always localized on a SB structure while a two state SACASSCF averages the two mirror related SB structures and sorts out solutions of u/g symmetry. The different results (D∞h or C∞v) obtained above by the same computational methods are mainly due to the initial asymmetrical geometry considered for the geometry optimization. When Δr = 0.1 bohr, the final optimized geometry was of D∞h character while when Δr = 0.3 bohr, the solution obtained was of C∞v nature. That explains the disagreement between the MRCI results in Refs. 14 and 17.
34.It is interesting to notice that in Ref. 17 the barrier ΔE(D∞h ← C∞v) diminishes when the level of the correlation treatment increases. By extrapolating their results, they may even obtain a zero barrier, i.e., no SB. By ameliorating the quality of the wavefunction, they partly restore the permutational symmetry.
35.A resonating CC wavefunction would eliminate the spurious double minimum, see, e.g., S. Yamanaka, S. Nishihara, K. Nakata, Y. Yonezawa, M. Okumura, T. Takada, H. Nakamura, and K. Yamaguchi, Int. J. Quantum Chem.109, 3811 (2009).
36.Towards that direction is the interesting work described in G. E. Scuseria, C. A. Jiménez-Hoyos, T. M. Henderson, K. Samanta, and J. K. Ellis, J. Chem. Phys.135, 124108 (2011), where they deliberately break any kind of symmetry and then by applying all suitable projection operators they restore it piece by piece.
37.To the best of our knowledge, the imposition of symmetry and equivalence restrictions to the HF orbitals is described for the first time in R. K. Nesbet, Proc. R. Soc. London, Ser. A230, 312 (1955);