Experimental, semi-experimental and ab initio equilibrium structures

154 Pages·2017·4.04 MB·English

by  Jean Demaison

Molecular Physics
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Experimental, semi-experimental and ab initio equilibrium
estructures
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Journal: Molecular Physics
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Manuscript ID: TMPH-2007-0302
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Manuscript Type: Invited Article
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Date Submitted by the i
11-Oct-2007
Author:
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Complete List of Authors: Demaison, Jean; Université de Lille I, PhLAM
Keywords: equilbrium structure, ab initwio, spectroscopy

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Page 1 of 152 Molecular Physics
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20 Experimentarl, semi-experimental and ab initio equilibrium structures
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23 P
J. Demaison
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28 Laboratoire de Physique des Laserrs, Atomes, et Molécules, UMR CNRS 8523, Université de
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30 Lille I, F-59655 Villeneuve d'Ascq Cédex, France
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34 E-mail: [email protected] ; fax: [phone]
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Number of pages: @
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Number of tables: 21
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Molecular Physics Page 2 of 152
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3 Abstract
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5 The determination of equilibrium structures of molecules by spectroscopic methods or
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7 by quantum mechanical calculations is reviewed. The following structures are described in
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detail: experimental equilibrium structures, empirical structures, semi-experimental structures
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and ab initio structures. The approximations made by the different methods are discussed and
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12 their accuracies are compared.
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1. Introduction
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19 The concept of molecular structure was developed during the 19th century. Couper [1],
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21 around 1856, was one of the first, with Kekulé [2], to state that carbon atom is tetravalent and
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23 to form a concrete idea Pof molecular structure. In 1885, von Baeyer [3] published his theory
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of ring strain whose main conclusion is that bond angles may deviate from the tetrahedral
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value of 109°28'. In 1874, the concept of tetrahedral carbon was simulteanously introduced by
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28 le Bel [4]and van't Hoff [5]. r
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30 The golden age of molecular structure research is the first half of the 20th century,
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perhaps starting with the discovery of X-ray diffraction around 1900 permitting to determine
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33 interatomic distances in crystals [6]. The setructure of diamond was obtained in 1913 by the
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35 Bragg's, father and son [7]. Debye [8] was thev first to determine the experimental structure of
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37 a molecule (CCl ) in the gas phase by X-ray diffraction. At about the same time, around 1930,
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the first gas-phase electron diffraction experiment was performed by Mark and Wierl [9].
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Although the accuracy of the measurements was not high, many molecules were studied in a
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relatively short time and the results were used to develop concepts concerning chemical
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44 bonding by Pauling [10] and others.
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46 In quantum chemistry, the first successful attempt to ncalculate the structure of a
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molecule was by Burrau in 1927 on H + [11]. It was followed, in the same year, by the
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y
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calculation of the bond length in H by Heitler and London [12].
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51 After the second world war microwave spectroscopy had arrived [13] which is
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53 generally considered to be the most precise technique for obtaining molecular geometries in
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the gas phase. This spectroscopy (as well as high resolution infrared spectroscopy) now
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determines rotational constants with a precision close to 1 in 108. If there was a simple
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58 relation between experimental rotational constants and equilibrium geometry, it would be
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60 possible to determine the structure of molecules with a tremendous accuracy. However, we
will see that many factors limit this accuracy.
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Page 3 of 152 Molecular Physics
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3 The apparition of the computer and its considerable increase in power are also
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5 extremely important because it permitted the development of ab initio electronic structure
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7 methods which can now provide accurate estimates of equilibrium structures.
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There are many reviews devoted to structure determination. However, most of the
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time, they limit themselves either to a single experimental technique or to ab initio
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12 calculations. There are relatively few papers where a critical comparison of the experimental
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14 and ab initio techniques is made and where the interplay of these methods is emphasized.
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Furthermore, although the techniques are described in great detail, there is no thorough
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discussion of the accuracy really achievable.
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19 The first question to answer is which accuracy is desirable. There are three reasons
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21 militing in favour of a high accuracy:
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23 • Theoreticians nePed accurate structures to check their more and more sophisticated
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computations. Although the accuracy of ab initio calculations varies wildly, it is
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reasonable to define a range from 0.3 pm to better than 0.1 pm.
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28 • Inspection of the range of ra few X-Y bond lengths shows that it is rather small. For
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30 instance, it is only 6 pm for the CH and NH bonds in different molecules. Thus, to
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compare a particular structure in different molecules, an accuracy significantly better
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33 than 1 pm is required. e
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35 • The energy of a molecule is sensitive vto its structure. Molecular mechanics programs
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37 for the calculation of properties of large molecules are parametrized against a small
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number of small molecules whose structure is assumed to be accurate. For instance,
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40 distortion of a C-C single bond by 2 pm "costws" 0.6 kJ mol-1; distortion of a ∠(CCC)
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42 bond angle by 2°, about 0.4 kJ mol-1; while the torsional distortion of a CCCC chain
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44 by 5° costs about 0.2 kJ mol-1 [14]. In order to be ablOe to determine the relative energy
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46 of a molecule with an acceptable accuracy (a few kJ/moln), it is necessary to scale the
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molecular mechanics programs [15] with the aid of molecules whose molecular
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geometry is very accurately known.
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51 The next step is to see under which conditions such an accuracy can be achieved. For this
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53 goal, we will first analyze the approximations which are made during the derivation of the
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55 molecular Hamiltonian.
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60 2. The molecular Hamiltonian [16]
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Molecular Physics Page 4 of 152
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3 A molecule is a collection of nuclei and electrons, held together by certain forces and
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5 obeying the laws of quantum mechanics. The dominant forces are the Coulomb electrostatic
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7 forces. Gravitational forces are also present but a simple calculation shows that they are
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completely negligible compared to the electrostatic forces. For open-shell molecules or when
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heavy nuclei are present, the electron spin magnetic moment may interact with other magnetic
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12 moments (generated by the orbital motion of the electrons or of the nuclei, or magnetic
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14 moments of the other electrons). These interactions are much smaller and will be neglected in
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order to simplify the presentation. When they are not negligible, they can be taken into
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17 account without difficulty1. What is left is the Coulomb Hamiltonian. To write it, it is
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19 assumed that the nuclei are point masses and the relativistic effects are negligible. The first
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21 approximation is a ve ry good one and will be further discussed in section 5.2. The second
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23 approximation is also a Psound one. However, when heavy nuclei are present or during highly
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accurate electronic structure computations, relativistic effects become important and should
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be taken into account. This approximation will be further discussed in section 3.5. Using these
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28 approximations, the molecular Hamriltonian may be written as
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R
32 H =TN+(Te+VNe+VNN+Vee)=TN+He (1)
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N
3345 with TN =−h2∑2ΔMα v (2a)
36 € α=1 α i
3378 T =−h2∑n Δi e (2b)
e m 2
39 € i=1
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N n
41 1 Z
V =− ∑∑ α (2c)
42 Ne 4πε r
43 € 0α=1i=1 iα
O
44 1 N n Z Z
45 V = ∑∑ α β (2d)
NN 4πε R
46 € 0α=1β>α αβ n
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48 1 n n 1
V = ∑∑ y (2e)
49 ee 4πε r
50 € 0 i=1 j>i ij
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∆ is the Laplace operator, M and Z are the mass and atomic number of the nucleus α, m is
52 α α
5534 € the mass of the electron, riα is the distance between electron i and nucleus α, and similar
55 definitions hold for r and R .
ij αβ
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60 1 The interactions of the magnetic and electric moments of the nuclei with the other electric and magnetic
moments in the molecule may lead to a splitting of the energy levels in several hyperfine components but they do
not affect the following discussion.
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Page 5 of 152 Molecular Physics
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3 2.1. The Born-Oppenheimer approximation and the electronic Hamiltonian [17]
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5 This Hamiltonian is too complicated to be solved exactly. As proposed by Born and
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7 Oppenheimer (BO), the nuclear kinetic energy T is first separated when motion of the
N
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electrons is investigated. The justification of the BO approximation is that the heavy nuclei
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move much more slowly than the light electons (it also assumes that the momentum of the
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12 electrons and the nuclei is of the same order of magnitude), thus T is an almost constant term
N
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14 and can therefore be neglected in the differential equation. In the remaining electronic
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Hamiltonian H , the nuclear positions R enter as parameters
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17 H ψ(e) = EBOψ(e) (3)
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19 Varying the position of R in small steps, one obtains EBO as a function of R. This is the
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21 potential energy (hype r)surface (PES) : EBO(R). Its minimum corresponds to the equilibrium
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23 structure of the moleculeP. As the nuclear masses are absent in He, EBO is isotopically invariant.
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25 In other words, in this approeximation all the isotopologues of an individual molecule have the
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same PES and the same equilibrium structure irrespective of the coordinate system used to
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28 represent R. It can be shown thrat the BO approximation can be trusted when the PESs
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30 corresponding to the different electronic states are well separated:
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32 E 0BO<<E1BO<<E2BO<<L (4)
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This is generally a good approximation for ground electronic states except for a few
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molecules. For instance, the first two excited electronic states of NO are close to the ground
36 € i 3
37 state and all three levels interact via vibronic couplings [18].
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39 The PES and the equilibrium structure can be calculated at different levels of electronic
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41 structure theory. This will be discussed in sections 3 and 6 together with the relativistic effects

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and the correction to the BO approximation.
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46 2.2. Vibration-rotation Hamiltonian [19] n
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48 In the second step of the BO approximation the nuclear kinetic energy T is
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reintroduced and the Hamiltonian is written as
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[T + EBO(R)]φ(R) (5)
52 N
53 where EBO(R) may be considered as the potential.
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55 The kinetic part of the vibration-rotation Hamiltonian was first derived by Wilson and
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57 Howard [20]. However, this original Hamiltonian is rather complicated (particularly due to
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the non-commutation of the various terms). Watson [21] could considerably simplify it. This
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led him to introduce a small correction term U which acts as a small mass-dependent effective
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Molecular Physics Page 6 of 152
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3 contribution to the potential energy. It comprises the translation of the center of mass
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5 described by three degrees of freedom (dof), the overall rotation of the molecule (three dofs)
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7 and the nuclear vibrations (3n – 6 dofs). The separation of the translational motion is
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straightforward but the (approximate) separation of vibrational and rotational motion is not a
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trivial problem. It is treated in many textbooks [16, 19, 20].
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12 After separation of rotation and vibration, a rotational Hamiltonian may be written for each
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14 vibrational state v as
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16 Hv = ∑BξJ2 +H , (6)
R Fv ξ cd
17 ξ=a,b,c
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19 where B ξv is the rootational constant in vibrational state v about the principal axis ξ, J ξ2 is the
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21 € component of the rotat ional angular momentum along the principal axis ξ and Hcd is a small
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23 €c orrection term called cPentrifugal distortion which is due to the fact that the€ molecule is not
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rigid and thus, upon rotation, the molecule is distorted by the stretching effects of the
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26 centrifugal forces.
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28 In a perturbational treatment, the rortational constant Bξ is given by
v
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30  d   d  d 
31 Bξv =Beξ −∑αξi  vi+ 2i +∑γξij vi+ 2Ri vj+ 2ji +L (7)
32 i i≥j €
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The summations are over all vibrational states, each characterized by a quantum number v
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3356 € and a degeneracy di. The parameters α ξi anvd i γ ξij are called vibration-rotation interaction
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constants of different order. Bξ is the equilibrium rotational constant. It is proportional to the
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39 inverse of the equilibrium mom€en t of in€er tia which is itself a function of the coordinates at
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41 equilibrium,
€
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43 h2
Bξ = O (8)
44 e 8π2Iξ(r )
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46 The convergence of the series expansion is usually fast, αξ nbeing about two orders of
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48 € magnitude smaller than Bξ and γξ two orders of magnitude smaller than αξ, see Table 1.
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50 The original rotational Hamiltonian cannot b€e used to analyze rovibrational spectra
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52 because it conta€i ns ma€n y terms which are not all independent€. Watson [22] submitted the
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54 Hamiltonian to unitary transformations which allowed him to reduce the number of
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parameters. The rotational constants are only marginally affected by the transformation. The
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rotational constants of an asymmetric top obtained from a fit using the Hamiltonian of Watson
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59 are affected by a small centrifugal distortion contribution which depends on the choice of the
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Page 7 of 152 Molecular Physics
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3 reduction and of the representation. Watson has shown that only the following linear
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5 combinations can be determined from the analysis of the spectra:
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B = B(A) +2Δ = B(S) +2D +6d (9a)
9 z z J z J 2
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B = B(A) +2Δ + Δ −2δ −2δ = B(S) +2D + D +2d +4d (9b)
11 x x J JK J K x J JK 1 2
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B = B(A) +2Δ + Δ +2δ +2δ = B(S) +2D + D −2d + 4d . (9c)
13 y y J JK J K y J JK 1 2
14
In eqs.(9) B(A) are the experimental constants in the so-called A reduction, B(S) are the
15 ξ ξ
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17 experimentalF constants in the so-called S reduction, and B are the determinable constants
ξ
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19 (wh€e re ξ = x, y, oz). However, these latter constants are still contaminate€d by the centrifugal
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21 distortion. As shown b y Kivelson and Wilson [23], the rigid rotor constants B ' are given by
ξ
22
1 1
23 B′ =B + (τ +τP+τ )+ τ
x x 2 yyzz xyxy xzxz 4 yzyz (10)
24
25 e €
26 where B′ and B′ are obtained by cyclic permutation of x, y, and z.
y z e
27 €
28 Finally, Watson has shown that thre rotational constants are affected by the mass-dependent
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contribution U to the potential energy. This contribution may be written as
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1 R
32 Brigid =B′ − [τ +τ +τ ] (11)
33 x x 8 xxxx xxyy xxzz e
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The problem is that the τ constants are experimentally determinable only for a planar
35 v
36 € molecule by means of the planarity relations of Diowling [24]. For a non-planar molecule they
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38 can be calculated from the harmonic force field [25e].
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These equations are approximate for three reasons: i) the result is sensitive to the particular
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choice of the rotational constants (equilibrium, ground state experimental, ground state
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43 corrected, etc.) used in the calculations ii) the higher-order terms (sextic terms) are neglected
O
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45 iii) there are still additional terms as shown by Chung and Parker [26] but whose expressions
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47 are not known to date. However, as noted by Aliev and Watson [27]l these last corrections are
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indistinguishable from the effects of the breakdown of the BO approxiymations.
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Compared to the other corrections, the centrifugal distortion correction is generally
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52 quite small except for very light molecules, see Table 2. Furthermore, it is different from zero
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54 only for asymmetric top molecules. However, in this case, it generally remains much larger
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than the experimental accuracy. In many practical cases, the spectroscopist will usually fit the
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line frequencies measured in order to obtain what he takes to be the experimental ground state
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59 rotational constants plus all quartic (for sufficiently high J also the sextic) centrifugal
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distortion constants. He then uses these rotational constants to determine the molecular
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Molecular Physics Page 8 of 152
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3 structure. It should be better to use the determinable constants, or still better to use the ground
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5 state rigid rotor corrected constants of Eq. (11). The spectroscopist seldom goes farther than
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7 to the determinable constants.
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2.3. Magnetic correction [28]
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12 Since the electrons tend to follow the motion of the nuclei, the bulk of the electronic
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14 contribution to the rotational constants can be taken into account by employing atomic rather
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than nuclear masses. This is a very good approximation for most molecules and it is about the
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only feasible one for polyatomic molecules. However, a small correction for unequal sharing
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19 of the electrons by the atoms and for nonspherical distribution of the electronic clouds around
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21 the atoms is sometimes nonnegligible and has to be taken into account.
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23 The total angular momePntum J of a molecule may be written as the sum of N, the angular
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momentum due to the rotation of the nuclei and L, the angular momentum of the electrons.
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The rotational Hamiltonian for the nuclear system plus the Hamiltonian for the unperturbed
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28 electronic energies may be written ras
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333012 H= 12∑ξ NIξξ2 +He = 12∑ξ (Jξ −IξLξ)2 +RHe
33 1 J2 J L 1 L2 e (12)
34 = ∑ ξ +H −∑ ξ ξ + ∑ ξ
35 2 I e I 2 I v
ξ ξ ξ ξ ξ ξ
36 14 4 2 4 4 3 14 2 4 3 i
37 H0=HR+He H ′
38 Since L is very small, the third term can be negleceted and H' can be treated as a perturbation
39 ξ
40 € of H0. We now assume that the molecule is not inw a pure 1Σ state ψ(0) (L = 0) but in a
41 0

42 perturbed state ψ(1) which has some electronic momentum. The correct effective rotational
0
43
O
44 Hamiltonian is then €
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46 H€ = ψ(1) H +H′ ψ(1) n (13)
47 eff 0 R 0 l
48 A simple perturbation calculation up to second-order gives
y
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50 €  2
555123 Heff = 12∑ξ Jξ2 I1ξ − I2ξ2 ∑n≠0 nEnL−ξE00  (14)

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55 This is equivalent to the definition of an effective moment of inertia (Iξ)eff by
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57 € nL 0 2
58 1 = 1 − 2 ∑ ξ (15)
59 (Iξ)eff Iξ Iξ2 n≠0 En−E0
60
€
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Page 9 of 152 Molecular Physics
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3 where I on the right is calculated using the nuclear masses. This effective moment of inertia
4 ξ
5 can be expressed as a function of the molecular rotational g factor in the principal axis system,
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7 whose definition is
8
2
910 g = Mp ∑Z (y2+z2)−2Mp ∑ nLx 0 (16)
11 xx Ix i i i i mIx n≠0 En−E0
12
and g and g are obtained by cyclic permutation. In this equation M is the mass of the
13 yy zz p
14
proton.
15 €
16 The effective rotational constant B (obtained from the analysis of the rotational spectrum) is
F eff
17
18 therefore
o
19
20 (Bξ) =Bξ + mrg Bn, (17)
21 eff M ξξ
p
22
23 where Bξ is the rotatioPnal constant calculated with atomic masses and Bn the rotational
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25 € constant calculated with nucelear masses.
26
The g factor can be obtained eexperimentally from the analysis of the Zeeman effect on the
27
28 r
rotational spectrum [28, 29]. It can also be calculated ab initio [30]. A few typical results are
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30 given in Table 3. As expected, the correction is the largest for very light molecules (as LiH)
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R
32 and it rapidly decreases when the mass of the molecule increases. There are, however, a few
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exceptions. As the expression of g shows, see Eq. (16), g may become large when an
34
35 v
electronic excited state is close to the ground state, (because the denominator E – E is
36 i n 0
37 small). This is the case for ozone (O ), where the magnetic correction is extremely large.
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