Molecular Spectra and Molecular Structure - Lec 9S

Lecture # 9 Supplement Excerpts from The Spectra and Dynamics of Diatomic Molecules 1 Chapter 3 Terms Neglected in the Born-Oppenheimer Approximation Contents 3.1 The Born-Oppenheimer Approximation . . . . . . 89 3.1.1 Potential Energy Curves . . . . . . . . . . . . . . . . 90 3.1.2 Terms Neglected in the Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . 92 3.1.2.1 Electrostatic and Nonadiabatic Part of H . 92 3.1.2.1.1 Crossing or Diabatic Curves . . . 93 3.1.2.1.2 Noncrossing or Adiabatic Curves . 94 3.1.2.2 The Spin Part of H . . . . . . . . . . . . . 94 3.1.2.3 Rotational Part of H . . . . . . . . . . . . 96 3.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . 99 3.2.1 Hund’s Cases . . . . . . . . . . . . . . . . . . . . . . 101 3.2.1.1 Definition of Basis Sets . . . . . . . . . . . 103 3.2.1.2 Quantum Numbers, Level Patterns, and the Effects of Terms Excluded from H(0) . . 113 3.2.1.3 Intermediate Case Situations . . . . . . . . 126 3.2.1.3.1 Introduction . . . . . . . . . . . . 126 3.2.1.3.2 Examples . . . . . . . . . . . . . . 127 3.2.1.4 Transformations Between Hund’s Case Ba­ sis Sets . . . . . . . . . . . . . . . . . . . . 130 3.2.1.5 Spectroscopic vs. Dynamical Hund’s Cases 136 3.2.1.6 Relationship between Noncommuting Terms in H and the Most Appropriate Hund’s Case137 3.2.2 Symmetry Properties . . . . . . . . . . . . . . . . . . 138 88 Terms Neglected in the Born-Oppenheimer Approximation 3.3 3.4 3.2.2.1 Symmetry Properties of Hund’s Case (a) Basis Functions . . . . . . . . . . . . . . . 138 3.2.2.2 Symmetry Properties of non-Hund’s Case (a) Basis Functions . . . . . . . . . . . . . 146 3.2.3 Molecular Electronic Wavefunctions . . . . . . . . . 148 3.2.4 Matrix Elements between Electronic Wavefunctions Electrostatic Perturbations . . . . . . . . . . . . . 156 161 3.3.1 Diabatic Curves . . . . . . . . . . . . . . . . . . . . 163 3.3.2 Approximate Representation of the Diabatic Elec­ tronic Wavefunction . . . . . . . . . . . . . . . . . . 165 3.3.3 Adiabatic Curves . . . . . . . . . . . . . . . . . . . . 168 3.3.4 Choice between the Diabatic and Adiabatic Models . 172 3.3.5 Electromagnetic Field-Dressed Diabatic and Adia­ batic Potential Energy Curves . . . . . . . . . . . . 177 Spin Part of the Hamiltonian . . . . . . . . . . . . 180 3.4.1 The Spin-Orbit Operator . . . . . . . . . . . . . . . 181 3.4.2 Expression of Spin-Orbit Matrix Elements in Terms of One-Electron Molecular Spin-Orbit Parameters . 183 3.4.2.1 Matrix Elements of the lzi · szi Term 3.4.2.1.1 . . . 183 Diagonal Matrix Elements . . . . 183 3.4.2.1.2 3.4.2.2 3.4.3 The Spin-Rotation Operator 3.4.4 The Spin-Spin Operator . . . . . . . . . . . . . . . . 196 3.6 . . . . . . . . . . . . . 191 3.4.4.1 Diagonal Matrix Elements of HSS : Calcu­ lation of the Direct Spin-Spin Parameter . 196 3.4.4.2 Calculation of Second-Order Spin-Orbit Ef­ fects . . . . . . . . . . . . . . . . . . . . . . 198 3.4.4.3 3.4.5 3.5 Off-Diagonal Matrix Elements . . 187 � − � − + Part Matrix Elements of the l+ i si + li si of HSO . . . . . . . . . . . . . . . . . . . . 190 3.4.4.2.1 π 2 Configuration . . . . . . . . . . 201 3.4.4.2.2 π 3 π  (or π 3 π  and ππ  ) Configu­ rations . . . . . . . . . . . . . . . 201 3 Off-Diagonal Matrix Elements . . . . . . . 202 Tensorial Operators . . . . . . . . . . . . . . . . . . 203 Rotational Perturbations . . . . . . . . . . . . . . . 210 3.5.1 Spin-Electronic Homogeneous Perturbations . . . . . 210 3.5.2 The S-Uncoupling Operator . . . . . . . . . . . . . . 212 3.5.3 The L-Uncoupling Operator . . . . . . . . . . . . . . 213 3.5.4 2 Π ∼ 2 Σ+ Interaction . . . . . . . . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . 227 3.1 The Born-Oppenheimer Approximation 3.1 89 The Born-Oppenheimer Approximation The exact Hamiltonian H for a diatomic molecule, with the electronic coordi­ nates expressed in the molecule-fixed axis system, is rather difficult to derive. Bunker (1968) provides a detailed derivation as well as a review of the coordi­ nate conventions, implicit approximations, and errors in previous discussions of the exact diatomic molecule Hamiltonian. Our goal is to find the exact solutions, ψiT (T = Total), of the Schrödinger equation, HψiT = EiT ψiT , (3.1.1) which correspond to the observed (exact) EiT energy levels. H is the nonrela­ tivistic Hamiltonian, which may be approximated by a sum of three operators, H = TN (R, θ, φ) + Te (r) + V (r, R), (3.1.2) where TN is the nuclear kinetic energy, Te is the electron kinetic energy, V is the electrostatic potential energy for the nuclei and electrons (including e− −e− , e− − N and N − N interactions), R is the internuclear distance, θ and φ specify the orientation of the internuclear axis (molecule-fixed coordinate system) relative to the laboratory coordinate system (see Section 2.3.3 and Fig. 2.4), and r represents all electron coordinates in the molecule-fixed system. The nuclear kinetic energy operator is given by � � � � � � ∂ −�2 1 ∂ ∂ 1 ∂2 2 ∂ R TN (R, θ, φ) = , + sin θ + 2µR2 ∂R ∂R sin θ ∂θ ∂θ sin2 θ ∂φ2 (3.1.3a) where µ= M A MB M A + MB is the nuclear reduced mass, with MA and MB the masses of atoms A and B. TN can be divided into vibrational and rotational terms, TN (R, θ, φ) = TN (R) + HROT (R, θ, φ). (3.1.3b) The electron kinetic energy operator is Te (r) = −�2 2m ∇2i , (3.1.3c) i where m is the electron mass and the summation is over all n electrons. † † The part of Te , � −�2 ∇i ∇j , 2(MA + MB ) i,j the diagonal matrix elements of which contribute, for example, 38 cm−1 for the ground state of H2 and 27 cm−1 for the B 1 Σ+ u excited state of H2 (Bunker, 1968), is neglected here. 90 Terms Neglected in the Born-Oppenheimer Approximation To solve Eq. (3.1.1), it would be useful to write the total energy as a sum of contributions from interactions between different particles. In decreasing order of importance, there are: electronic energy, E el , vibrational energy, G(v), and rotational energy, F (J). In fact, this separation is assumed whenever the expression T = E el + G(v) + F (J) Eapp (3.1.4) is used to represent observed energy levels. However, Eq. (3.1.4) is always an approximation. It is never possible to express E T exactly as in Eq. (3.1.4). This means that it is not possible to separate H rigorously into terms corresponding to the different motions of the particles. The approximate wavefunction sug­ gested by the desired but approximate energy expression, (Eq. (3.1.4)), is a product of two functions, BO = Φi,Λ,S,Σ (r; R)χv (R, θ, φ), ψi,v (3.1.5) where the first factor is the electronic wavefunction and the second is the vibration-rotation wavefunction. Λ is the projection of the electronic orbital angular momentum on the internuclear axis, S is the spin angular momentum, and Σ is its projection on the internuclear axis. The approximate solution [Eq. (3.1.5)] is called a Born-Oppenheimer (BO) product function. It corresponds to a solution where all of the couplings in H between nuclear and electronic motions are ignored. 3.1.1 Potential Energy Curves Since TN is smaller than Te by the factor m/µ, it can be neglected initially. Φ is then the solution of the clamped nuclei electronic Schrödinger equation, [Te (r) + V (r, R)]Φi (r; R) = Eiel (R)Φi (r; R) (3.1.6) or Hel Φi = Eiel Φi . If the approximate product function, Φi χ, is inserted into Eq. (3.1.1), after multiplying by Φ∗i and integrating over the electronic coordinates r, one obtains � � Φi |TN + Te + V |Φi r χ = E T Φi |Φi r χ. The electronic wavefunction is normalized to unity because it is a probability distribution function, Φi |Φi r = 1. If the effect of ∂/∂R [contained in TN (R)] on the electronic wavefunction, which certainly depends on R, is neglected, then � � Φi |TN |Φi r χ  TN Φi |Φi r χ = TN χ. Thus, Eq. (3.1.6) may be simplified to Φi |Te + V |Φi r = Eiel (R) 3.1 The Born-Oppenheimer Approximation 91 and the result used to obtain the nuclear Schrödinger equation, [TN (R, θ, φ) + Eiel (R)]χ(R, θ, φ) = E T χ(R, θ, φ). Since TN (R, θ, φ) = TN (R) + HROT , the radial and angular variables can be separated, as in the case of the hydrogen atom (Pauling and Wilson, 1935), as J χ(R, θ, φ) = χv,J (R)DΩM (α = π/2, β = θ, γ = φ) = χv,J (R) αβγ|JM Ω , where αβγ|JM Ω is the symmetric rotor function defined by Eq. (2.3.41) and χv,J is a vibrational eigenfunction of � N � T (R) + (�2 /2µR2 )[J(J + 1) − Ω2 ] + Eiel (R) χv,J (R) = E T χv,J (R), (3.1.7) where Ω = Λ + Σ. Eiel may be viewed as the potential energy curve in which the nuclei move, but it must be emphasized that potential energy curves do not correspond to any physical observable. They are a concept, derived from a specified set of assumptions for defining a particular type of approximate wavefunctions [Eq. (3.1.5)]. The observed levels are not exact energy eigenvalues of a given potential curve. In general, the separation between the electronic and nuclear motions, which constitutes the BO approximation, is convenient. But when the observed levels do not fit formulas such as Eq. (3.1.4), it is simply because the function [Eq. (3.1.5)] is a bad approximation in that particular case. An exact solution, ψiT , of the total Hamiltonian H must satisfy two condi­ tions: � T � T T (i) �ψiT |H|ψiT � = Ei (3.1.8) for any j =  i. ψj |H|ψi = 0, (ii) � � One then says that the Hamiltonian is diagonalized in the basis set ψ T . In principle, it is possible to express any exact solution as an infinite expan­ sion over the BO product functions, ψiT = ∞ � ci,vj ΦBO j χvj . (3.1.9) j,vj The coefficients of this expansion are determined by diagonalizing a ma­ trix representation of the total Hamiltonian, constructed by evaluating matrix elements between BO basis functions. It is most useful to define a basis set of the type of Eq. (3.1.5) for which the off-diagonal matrix elements of the total Hamiltonian are as small as possible. If one term of the expansion in Eq. (3.1.9) is sufficient, this means that the BO approximation is valid. Fortunately, when the BO approximation fails, often only two terms of the BO expansion are sufficient. If large off-diagonal matrix elements of the Hamiltonian exist that couple many vibrational wavefunctions belonging to two different electronic states, then 92 Terms Neglected in the Born-Oppenheimer Approximation it is much more convenient to write and solve the coupled differential equations that describe these two states (see Section 4.4.3). The BO approximation fails whenever off-diagonal elements of H, Hij , connecting different eigenstates of the Eq. (3.1.5) form, are large compared to the difference between the diagonal elements, Hii − Hjj . 3.1.2 Terms Neglected in the Born-Oppenheimer Approximation This section deals with various types of nonzero off-diagonal matrix elements of H between approximate BO product basis functions. In order to go beyond the BO approximation, to try to obtain an exact solution, it is necessary to use a BO representation. In other words, the exact eigenvalues and eigenfunctions, which can be compared to observed energy levels, are expressed in terms of the BO † representation, as a linear combination of BO product functions. � BOspecifically � Presently, ψ is the only available type of complete, rigorously definable basis set. In the following, the off-diagonal matrix elements of H = Hel + TN(R) + HROT of the form � Φi,Λ,S,Σ χvi ,J |H|Φj,Λ ,S  ,Σ χvj ,J � will be discussed. The off-diagonal matrix elements of Hel give rise to electro­ static perturbations. The off-diagonal matrix elements of TN(R) give rise to nonadiabatic interactions. The off-diagonal matrix elements of HROT give rise to rotational perturbations. The total Hamiltonian discussed above does not include the relativistic part of the Hamiltonian. That contribution to H will be introduced as a phenomenological perturbation operator, HSO , and will give rise to spin-orbit perturbations. 3.1.2.1 Electrostatic and Nonadiabatic Part of H In Section 3.3 it will be shown that, to describe perturbations which result from neglected terms in the Hel + T N (R) part of the Hamiltonian, two differ­ ent types of BO representations are useful. If a crossing (diabatic) potential curve representation is used, off-diagonal matrix elements of Hel appear be­ tween the states of this representation. If a noncrossing (adiabatic) potential curve representation is the starting point, the TN operator becomes responsible for perturbations. † It is never appropriate to take linear combinations of vibrational basis states which belong to different potential curves unless the associated electronic and rotational factors are included. 3.1 The Born-Oppenheimer Approximation 3.1.2.1.1 93 Crossing or Diabatic Curves If Φi and Φj are two different exact solutions of Eq. (3.1.6), then � � Φi |Hel |Φj = 0 It will be shown later that exact solutions of the electronic Schrödinger equation can give rise to double minimum potential curves. Such potentials can be incon­ venient for treating some perturbation situations. It is often more convenient to start from approximate solutions of Hel where potential curves, which would have avoided crossing for the exact Hel , actually cross. In such a case, � app el app � e (R) = 0. (3.1.10) Φi |H |Φj r = Hi,j The expression for Hel includes Te (r) [Eq. (3.1.3c)] and V (r, R), where V (r, R) = V eN (r, R) + V ee (r) + V N N (R); V eN (r, R) is the Coulomb electron-nuclear attraction energy operator, � n � � ZA e2 ZB e2 V eN (r, R) = − ; + rAi rBi i=1 V ee (r) is the Coulomb interelectronic repulsion energy operator, V ee (r) = n � i=1 e2 , r j>i ij where j > i ensures that each repulsion between the ith and jth electrons is considered only once; and V N N (R) is the Coulomb internuclear repulsion energy operator, V N N (R) = ZA ZB e2 /R. Note that the negative sign of V eN implies that it contributes to energy stabi­ lization. Crossing curves are obtained by excluding parts of the spin-orbit term, † HSO , and of the interelectronic term, V ee , from the Hel operator. The effect ee of V , discussed in Section 3.3.2, is extremely important as it compromises the validity of the single electronic configuration picture which is often taken as synonymous with the diabatic potential curve picture. In both crossing and noncrossing curve approaches, perturbations between levels of the same symmetry can occur. In the diabatic picture, these are usu­ ally called “electrostatic perturbations” because they arise from V ee . In the adiabatic picture, they arise from TN , the nuclear kinetic energy operator, but are often misleadingly called electrostatic perturbations. † It is not possible to give a unique definition of a diabatic potential curve without identi­ fying the specific term in Hel that is excluded. The impossibility of identifying such a term and the consequent nonuniqueness of the a priori definition of diabatic curves is discussed by Lewis and Hougen (1968), Smith (1969), and Mead and Truhlar (1982) (See also Section 3.3.2). Diabatic curves may be defined empirically (Section 3.3) by assuming a deperturbation model e (R) is independent of R or, at most, varies linearly with R]. [e.g., that Hi,j 94 3.1.2.1.2 Terms Neglected in the Born-Oppenheimer Approximation Noncrossing or Adiabatic Curves Equation (3.1.7) was obtained by assuming that TN (R) does not act on the electronic wavefunction. Actually, the Φi are functions of the nuclear coordinate, R. Adding to Eq. (3.1.7) the neglected R–dependent term � � Φi |TN |Φi r , where integration over all electronic coordinates and the θ, φ nuclear coordinates is implied, one obtains � N � � � T (R) + (�2 /2µR2 )[J(J + 1) − Ω2 ] + Φi |TN |Φi r + Eiel (R) χv,J (R) = E T χv,J (R). (3.1.11) The potential curves defined by � � Eiad (R) = Eiel (R) + Φi |TN |Φi r are called adiabatic potential curves, but the second term makes a much smaller contribution to the energy than E el (R) � (Bunker,�1968). The off-diagonal matrix element Φi |TN |Φj r , the nonadiabatic coupling term, is examined in Section 3.3.3. This type of matrix element appears between states of identical symmetry and gives rise to homogeneous perturbations. 3.1.2.2 The Spin Part of H Equation (3.1.2) is the nonrelativistic Hamiltonian. This means that the spindependent part of the Hamiltonian (HSO spin-orbit and HSS spin-spin) has been neglected. The electronic angular momentum quantum numbers, which are well-defined for eigenfunctions of nonrelativistic adiabatic and diabatic potential curves, are Λ, Σ, and S (and redundantly, Ω = Λ + Σ). Off-diagonal matrix elements, � � Φi,Λ,Σ,Ω |HSO + HSS |Φj,Λ ,Σ ,Ω , can be nonzero between states of different Λ and S (but identical Ω) quantum numbers, corresponding to different solutions [Eq. (3.1.5)] of the nonrelativistic Hamiltonian [Eq. (3.1.2)]. Matrix elements of this type are discussed in Sec­ tion 3.4. The spin-spin part, HSS , of the spin Hamiltonian usually gives rise to matrix elements much smaller than those of HSO and Hel . The relativistic Hamiltonian may be defined by adding HSO to Hel . The eigenfunctions of this new Hamiltonian are the relativistic wavefunctions, Φi,Ω , which define the relativistic potential curves � � Φi,Ω |{Hel (R) + HSO (R)}|Φi,Ω r = Eir (R), (3.1.12) where now the only good electronic angular momentum quantum number is Ω = Λ + Σ. Table 3.1 summarizes the different types of potential energy curves and the specific terms in H that are neglected in order to define the diabatic, adiabatic, relativistic-adiabatic, and relativistic-diabatic basis functions. 162 Terms Neglected in the Born-Oppenheimer Approximation Electrostatic valence∼Rydberg N2 b 1 Σ+ ∼ c 1 Σ + u (vc ) and u (vb ) b1 Πu (vb ) ∼c1 Πu (vc ) perturbations. Each segment of the absorption spectrum (from Yoshino, et al., 1979) shows several perturbing states near a nominal vc = 0 − 4 c 1 Σ+ u level. The b and b valence states are perturbed by the c and c Rydberg states (nominally the 1 Πu and + 1Σ u components of a 3p complex) as well as by higher Rydberg states. The ∆Gv+1/2 plot + state (from Dressler, 1969; see also Fig. 3.6) in the lower left corner shows the for the b 1 Σu massive level shifts that had made it difficult to recognize the electronic state parentage of the observed singlet vibronic levels of N2 . The largest positive deviations of the observed ∆G val­ ues from the smoothly varying deperturbed value (dashed line) occur when the vc = 1, 2 and 3 levels are sandwiched between the vb = 3 and 4, 6 and 7, and 9 and 10 levels, respectively. These b ∼c perturbations are discussed in Section 6.3.1 and further illustrated in Fig. 3.6. Figure 3.4: but then � N ad Φad 1 |T |Φ2 � r = 0. (3.3.3) In principle, whatever the initial model, after introducing the vibronic coupling terms corresponding to the chosen type of deperturbed potential curves, the experimental energy levels are obtained by diagonalizing one or the other type of interaction matrix. One example will be discussed later (Section 3.3.4). If the deperturbed curves intersect and are characterized by very different molecular constants, then they are diabatic curves (see Fig. 3.5). If the crossing is avoided, adiabatic curves are involved, and one of these curves can have a double minimum. One frequently finds that, in the region of an avoided crossing, the adiabatic wavefunction changes electronic character abruptly and the derivative of the electronic wavefunction with respect to R can be large. In 3.3 Electrostatic Perturbations 163 fact, it is this derivative that controls the size of the matrix elements of the TN operator [Eq. (3.3.10)]. In the older literature (Dieke, 1935; Kovács, 1969), there is some confusion. Only the TN operator has been assumed to connect states of the same symmetry, and the potential curves of these interacting states have been assumed to cross. Neither assumption is correct. 3.3.1 Diabatic Curves The vibronic interaction between the level v1 of the diabatic potential curve V1d (R) and the level v2 of another diabatic curve V2d (R) is reduced to � � H1,v1 ;2,v2 = Φd1 χdv1 |Hel |Φd2 χdv2 (3.3.4) since, by definition, � d N d� Φ1 |T |Φ2 = 0. In the diabatic model, the electronic part of the matrix element H1,v1 ;2,v2 is often assumed to be independent of R. Then the nuclear and electronic coordinates can be separated in the integration of Eq. (3.3.4). By integration over the electronic coordinates, one obtains � � H1,v1 ;2,v2 = H e v1d |v2d , (3.3.5) where � � H e = Φd1 |Hel |Φd2 and � � v1d |v2d = � χdv1∗ (R)χdv2 (R)dR. Indeed, as for any electronic quantity, the value of H e actually depends on R. However, this dependence is usually weak. Equation (3.3.5) holds even if the R– dependence of H e is a linear function of the R–centroid (Halevi, 1965) defined by R̄ij = vi |R|vj  . vi |vj  (3.3.6a) The significance of the R–centroid is illustrated as follows. The electronic matrix element, H e (R), may be expanded in a power series about an arbitrar­ ily chosen internuclear distance, R (most usefully, R = RC , the internuclear distance at which the two potential curves cross), 164 Terms Neglected in the Born-Oppenheimer Approximation Figure 3.5: Diabatic and adiabatic potential curves. The diabatic curves (solid lines) cross at Rc and are defined by neglecting the part of Hel that causes the adiabatic curves (dotted lines) to avoid crossing by 2H e at RC . e e  H (R) = H (R ) + � dH e dR � 1 (R − R ) + 2 R=R  � d2 H e dR2 � R=R (R − R )2 . (3.3.7a) Then the vibrational matrix elements of H e (R) are expressed in terms of Rn − centroids, n = Rij vi |Rn |vj  vi |vj  (3.3.6b) � dH e vi |H e (R)|vj  = H e (R ) vi |vj  [vi |R|vj  − R vi |vj ] dR R=R � � �� � � 1 d2 H e 2  2 + v | R | v v | R | v  + R  v | v  − 2R i j i j i j 2 dR2 R=R � � � dH e Rij − R dR  �R � 2 R= � � e 1 d H 2 − 2R R + R 2 . + Rij ij 2 2 dR R=R vi |H e (R)|vj  = H e (R ) + vi |vj  � (3.3.7b) Now, making the R–centroid approximation, which is quite distinct from the Rn –centroid expansion, namely, ¯ R n Rij n− Rij = vi |Rn |vj  vi |Rn−1 |vj  � �n [or, in other words, Rn = R ], then Eq. (3.3.7b) becomes � � vi |H e (R)|vj  dH e e  (Rij − R ) = H (R ) + vi |vj  dR R=R � � 1 d2 H e + (Rij − R )2 , 2 dR2 R=R 3.3 Electrostatic Perturbations 165 which is identical to Eq. (3.3.7a) where R is set equal to Rij . For near-degenerate vibrational levels of any two crossing potential curves, the R–centroid has the convenient property of being nearly equal to RC , the R–value where the two curves cross (Schamps, 1977). Thus, � R = Rk Rk−1 = RC , and, setting R = RC , vi |H e |vj  = H e (RC ). vi |vj  The R–centroid approximation has repeatedly been tested numerically. For perturbations between levels of potentials that intersect exactly once, the R– centroid approximation can be regarded as more accurate than experimentally measurable matrix elements. The validity of the R–centroid approximation is based on a stationary phase argument (see Section 5.1.1) (Tellinghuisen 1984). For two vibrational states, |vi  and |vj , at nearly identical total energy, the vibrational overlap integral I(R ) = � R vi |R R|vj  0 accumulates only near R = RC , where the two vibrational wavefunctions os­ cillate at the same spatial frequency. This stationary phase argument works equally well at low and high v and for RC near or far from vibrational turning points, provided that there is only one intersection between the potential curves within the range of R defined by the vibrational turning points at the total energy of the perturbation. Approximate deperturbed curves can be derived from unperturbed vibra­ tional levels far from the energy of the curve crossing region. The overlap factor between vibrational wavefunctions is calculable numerically. (Note that a Franck-Condon factor is the absolute magnitude squared of the overlap fac­ tor.) From Eq. (3.3.5) and the experimental value of H1,v1 ;2,v2 , an initial trial value for H e can be deduced. If the value of H e is as large as the value of ωe (see Table 4.4), then electrostatic interactions strongly perturb the entire set of vibrational levels. Figure 3.6 shows the irregular pattern of ∆G values for the perturbed 1 Σ+ u states of N2 . Final deperturbed curves are obtained by diagonalization of the matrix, including all vibrational levels of both states, as described in Section 4.4. 3.3.2 Approximate Representation of the Diabatic Electronic Wavefunction The single-configuration approximation is an approximate representation for the diabatic electronic function. Such an approximation is expected to result in 166 Terms Neglected in the Born-Oppenheimer Approximation Figure 3.6: Variation of ∆Gv for the mutually interacting b and c 1 Σ+ u states of N2 . The solid and dashed lines correspond, respectively, to the observed and calculated (LefebvreBrion, 1969) values. The deperturbed c (v = 2) and b (v = 7) levels are nearly degenerate and interact strongly (see Table 5.4). This accounts for the largest ∆G anomalies. smooth R–variation of the electronic wavefunction. In the case where the con­ figurations of the two interacting states differ by two spin-orbitals, the electronic matrix element of the TN operator between these two approximate functions is exactly zero, since it can be demonstrated that the TN operator acts as a oneelectron operator (Section 3.2.4) (Sidis and Lefebvre-Brion, 1971). However, as the electronic Hamiltonian, Hel , contains a two-electron operator, � e2 , r i 1011 W/cm2 electro­ magnetic fields typically provided by picosecond or femtosecond pulsed lasers, it is necessary to solve the time-dependent Schrödinger equation in a non­ perturbative approach (see Section 9.1.2). The total Hamiltonian is � � � ◦  H = H + E (t) · e ri (3.3.18) i ◦ where H is the field-free Hamiltonian of Eq. (3.1.1) and �E 0 cos ωt E(t) = k (3.3.19) 5.76 Lecture # 9 Supplement Page 2 Electronic Configurations It is usually possible and useful to assign observed molecular electronic states to electronic config­ urations. A configuration is a list of the molecular orbitals occupied by electrons. Molecular orbitals are usually specified by n�λ, where n is an integer > 1, � is the orbital angular momentum (� = 0, 1, 2, 3, 4 → s, p, d, f, g), and λ is the projection of � on the internuclear axis (λ ≤ �, λ = 0, 1, 2, 3, 4 → σ, π, δ, φ, γ). Two conventions exist for n, the molecular orbital convention attributes two sets of orbitals �λ and �λ∗ for each value of n (bonding and antibonding*), the other more common convention simply numbers the �λ orbitals in order of energy. Often (particularly for Rydberg states) n has the same meaning as the principal quantum number of atoms. The electron configuration is written by enclosing each orbital designation in parentheses and writing the occupancy as a right superscript. The X 1 Σ+ state of CO (14 electrons) belongs to the (1sσ)2 (1sσ∗ )2 (2sσ)2 (2sσ∗ )2 (2pσ)2 (2pπ)4 configuration. The maximum occupancy of a σ orbital is 2, of a π orbital 4, of a δ orbital 4, etc. Note that all orbitals for the CO X 1 Σ+ state are filled. Filled orbitals can give rise only to 1 Σ+ states. In working out molecular properties from 1-electron functions, the closed-shell core electrons are ignored. Note also that the molecular orbital picture describes CO as having a triple bond. Often configurations are abbreviated by omitting filled shells. Excited states of CO belong to the following configurations. (2pπ)4 (2pσ)(2pπ∗ ) (2pπ)3 (2pσ)2 (2pπ∗ ) A1 Π, a3 Π , a� 3 Σ+ , e3 Σ− , d3 Δ, I1 Σ− , D1 Δ, 1 Σ+ The reason L is undefined in molecules is illustrated by the above configuration labels: 2pσ � 2pπ, but in an atom (2pπ)(2pσ) → (np)2 which gives rise to states with L = 0, 1, 2 and S = 0, 1. Thus, whenever more than 1 � > 0 electron is outside of a closed shell, there will be an essential ambiguity about L. L can be evaluated using ab initio wave functions to calculate the expectation value of L2 . Usually, molecular states can be unambiguously assigned a configuration label. Although con­ figuration mixing is always present, it is usually relatively small (< 5%) and relatively independent of internuclear distance. In such cases, configuration labels are particularly useful in: 1. The analysis of perturbations – the magnitudes of perturbation interactions (AΠΣ L+ , BΠΣ L+ ) for various perturbations may be related to each other or even to AΠ and BΠ . 2. Lambda doubling may be related to either BΠ and AΠ (pure precession) or AΠΣ L+ , BΠΣ L+ . 3. Properties of Rydberg states may be related to those of ionic states. 4. Additional selection rules for observability of perturbations are obtained and may be used to assign configuration labels to other states. 5.76 Lecture # 9 Supplement Page 3 5. The sign of AΠ and AΠ,Σ may be predicted. A general method for construction of basis functions as products of 1-electron spin-orbitals is clearly described in The Theory of Atomic Spectra, pages 160-174, by E. U. Condon and G. H. Shortley. Wave functions which are antisymmetric with respect to interchange of electrons are compactly written as a determinant of 1-electron functions. The columns of the determinant correspond to the electron index, and the rows correspond to the occupied spin-orbitals. For example, a (2pπ)2 (2pσ) configuration has the following spin-orbitals (in standard order C & S, p. 169): |nλσ� = (2p1+ )(2p1− )(2p − 1+ )(2p − 1− )(2p0+ )(2p0− ) The ± right superscript refers to the electron spin ± 12 . To write a determinantal function corresponding to a particular state of the π2 σ configuration one picks the three spin-orbitals needed to obtain the desired Λ, Σ, Ω. The states derived from π2 σ are 2 Σ+ , 2 Σ− , 2 Δ, 4 Σ− . 2 1 Δ5/2 = √ 3! �� � ��2p1+ (1) 2p1+ (2) 2p1+ (3)��� �� � ��2p1− (1) 2p1− (2) 2p1− (3)��� �� �� �� � �2p0+ (1) 2p0+ (2) 2p0+ (3)�� Using a shorthand, we drop the 2p and write only the three diagonal functions. The procedure for assigning the various determinants to |ΛS Σ� basis functions is laborious. First write all determinants for each value of Λ and Σ. 1 2 Λ = 2, Σ= Λ = 2, Σ=− Λ = 0, Σ= 3 2 Λ = 0, Σ= 1 2 1 2 �1+ (1) 1− (2) 0+ (3)� 2 Δ3/2 �1+ (1) 1− (2) 0− (3)� 2 Δ1/2 �1+ (1) − 1+ (2) 0+ (3)� ⎧ + ⎫ ⎪ �1 (1) − 1+ (2) 0− (3)�⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ⎨ ⎬ − + �1 (1) − 1 (2) 0 (3)� ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩�1− (1) − 1+ (2) 0+ (3)�⎪ ⎭ 4 − Σ3/2 4 Σ1/2 , 2 Σ+ , 2 Σ− . Omit negative values of Λ. There is an ambiguity for Λ = 0, Σ = 12 . We use S− and σv to assign the 5.76 Lecture # 9 Supplement Page 4 remaining three functions to 2 Σ+ , 2 Σ− , 4 Σ−1/2 � � � � S− ��4 Σ−3/2 = [(S + Σ)(S − Σ + 1)]1/2 ��4 Σ1/2 � ��4 − � � �Σ = (3)−1/2 S ��4 Σ− 1/2 − S− = 3 � 3/2 si− . i=1 Thus 3 � ��4 − � � Σ1/2 = (3)−1/2 si− �1+ (1) − 1+ (2) 0+ (3)� i=1 = (3) −1/2 � − � �1 − 1+ 0+ � + �1+ − 1− 0+ � + �1+ − 1+ 0− � � �� �� ��1/2 � Since s− �� s = 12 , σ = 12 = 12 + 12 12 − 12 + 1 = 1. We know doublet functions must be orthogonal to quartet so try ⎧ − + + + − + ⎪ ⎪ ⎨ �1 − 1 0 � − �1 − 1 0 � ⎪ ⎪ ⎩ −�1− − 1+ 0+ � − �1+ − 1− 0+ � + 2�1+ − 1+ 0− � and observe effect of σv on these mutually orthogonal functions. σλv |�λ� = (−1)�−λ |� − λ� , � = 1 σΛv |LΛ� = ±(−1)L−Λ |L − Λ� , L ≡ 2 arbitrarily chosen � � � � σΛv ��2 Σ+ = + ��2 Σ+ � � � � σΛv ��2 Σ− = − ��2 Σ− σΛv = Πi σλvi σλv1 σλv2 σλv3 [�1− − 1+ 0+ � − �1+ − 1− 0+ �] = (−1)0+2+1 � − 1− 1+ 0+ � − (−1)0+2+1 � − 1+ 1− 0+ � = (−1)[� − 1− 1+ 0+ � − � − 1+ 1− 0+ �] but the two determinants are not in standard order. An interchange of adjacent columns results in a sign change. Thus σΛv [�1− − 1+ 0+ � − �1+ − 1− 0+ �] = [�1+ − 1− 0+ � − �1− − 1+ 0+ � = (−1)[�1− − 1+ 0+ � − �1+ − 1− 0+ �] 5.76 Lecture # 9 Supplement Page 5 Thus the function belongs to 2 Σ− . So 2 Σ−1/2 = 2−1/2 [�1− − 1+ 0+ � − �1+ − 1− 0+ �] and 2 Σ+1/2 = 6−1/2 [2�1+ − 1+ 0− � − �1− − 1+ 0+ � − �1+ − 1− 0+ �] So we have explicitly written out all the states of the (2pπ)2 (2pσ) configuration. Now consider the states of the (2 pπ)(2pσ)2 configuration. There is only a 2 Π state. 2 Π3/2 = �1+ 0+ 0− � 2 Π1/2 = �1− 0+ 0− �. How do we take matrix elements using these functions? The usual operators may be re-written in terms of 1-electron or 2-electron operators. � � � � 1 ·S≡ ai �iz siz + (�i+ si− + �i− si+ ) . AL a i � i · si = ∼ 2 i i · S has ΔS = 0, ±1 selection rule. Note that this form of AL ∼ ⎛ ⎞ ⎛ ⎞ ⎜⎜⎜� ⎟⎟⎟ ⎜⎜⎜� ⎟⎟⎟ s j ⎠⎟⎟ . BL · S = B ⎝⎜⎜ �i ⎠⎟⎟ · ⎝⎜⎜ i i Note that this form of BL · S has exclusively the ΔS = 0 selection rule. � J·L=J· �i J·S=J· i � si . i These operators should give exactly the same matrix elements we obtained using |ΩJM� |ΛS Σ� functions. � � Consider 2 Π1/2 |AL · S| 2 Σ−1/2 . A 1-electron operator can have nonzero matrix elements between product functions differing by no more than 1 spin-orbital. In addition, no 1-electron angular momentum operator can change n (principal quantum number). Thus we have a new selection rule for perturbations. This is what is meant by configurationally forbidden perturbations. Configurationally forbidden perturba­ tions are: 1. Δn � 0 2. Or involve states belonging to configurations differing by more than one orbital. 5.76 Lecture # 9 Supplement Page 6 How do we evaluate matrix elements? (C & S, p. 169-171). �� �� ��� ��� 3 � � � � � � � 1 2 Π1/2 |AL · S|2 Σ−1/2 = 2−1/2 ���1− 0+ 0− ��� ai �i+ si− ��� 1− − 1+ 0+ ��� � � i=1 2 � � �� �� ��� ��� 3 � � � � � 1 = −2−1/2 ���1− 0+ 1− ��� ai �i+ si− ��� 1+ − 1− 0+ ��� . � � i=1 2 � � The locations of the identical spin-orbitals in the left– and right–hand functions must agree and the pair of differing spin-orbitals must also be at the same location. The right–hand function in the first term must be rearranged by a switch of −1+ and 0+ to match the left–hand function. The second term cannot be rearranged because the left and right functions differ by two spin-orbitals. Thus � 2 Π1/2 |A L · S|2 Σ−1/2 ∼ � �� ��� �� ��� 3 � � � � � ai = −2−1/2 ���1− 0+ 1− ��� �i+ si− ��� 1− 0+ − 1+ ��� . � � i=1 2 � � The first two terms in the summation over i are zero, because the difference between the left and right functions is in the third spin-orbital and �+ s− causes a change in spin-orbital. � 2 � �� 2 − � � � � Σ1/2 = −2−3/2 0− |a�+ s− | − 1+ S Π1/2 ��AL · ∼ � � 1 1 −3/2 = −2 �0 |a�+ | − 1� − |s− | 2 2 �� �� ��1/2 1 1 1 1 −3/2 = −2 + − +1 �0 |a�+ | − 1� 2 2 2 2 = −2−3/2 �0 |a�+ | − 1� . Often, because � is usually not a good quantum number, only the s− part of the matrix element is evaluated. If � is good, then � � �� 2 − � 2 � Σ1/2 = −2−3/2 �0 |�+ | − 1� �n = 2|a|n = 2� S Π1/2 ��AL · ∼ = −2−3/2 [(1 + 1)(1 − 1 + 1)]1/2 �n = 2|a|n = 2� . n = 2 because we started with 2pπ, 2pσ orbitals. All matrix elements involving the (2 pπ) (2pσ)2 Π state and the (2 pπ)2 (2pσ) 2 Σ+ , 2 Σ− , 4 Σ− , 2 Δ states can be re-expressed in terms of matrix elements of the form 2 �n = 2, � = 1, λ = 1 |a�+ | n = 2, � = 1, λ = 0� B �n = 2, � = 1, λ = 1 |�+ | n = 2, � = 1, λ = 0� 5.76 Lecture # 9 Supplement Page 7 and thus the magnitudes of all such perturbations can be inter-related. If the �λ labels used to describe the configuration are not exact, these matrix elements indicate what the approximate value of � is for a valence or a Rydberg orbital. A Rydberg orbital is a diffuse atomic–like, nearly spherical orbital. Usually n and � are good quantum numbers for Rydberg orbitals. The nuclei and inner orbitals look a “fat proton” to an electron in a Rydberg orbital. The “fat proton” looks to the spectroscopist like one of the electronic states of the corresponding ion, in fact a molecule in a Rydberg state has B(v) and G(v) functions nearly identical to those of the ionic state at the Rydberg series limit. The Rydberg series limit corresponds to n = ∞. Rydberg states are usually specified by writing the state of the ionic core, followed by the quantum numbers of the Rydberg orbital. For example, in NO 2 + Σ (NO+ X 1 Σ+ )7sσ or 2 Π(NO+ A1 Π)5dσ. The states in a Rydberg series form a progression R (n − δ)2 where T ∞ is the series (ionization) limit and δ is the quantum defect. δ is usually small (less than 1) and is not quite constant. Note that Rydberg states have vibrational levels, but often only v = 0 is observed. A valence orbital is compact, and n and � are usually not good quantum numbers. A valence orbital is perfectly well specified by n, �, λ, but one should regard these quantum numbers as labels only. Valence orbitals are usually lower in energy than Rydberg orbitals and higher in energy than the core orbitals. Valence states usually have longer re and smaller ωe than either Rydberg states, ionic states, or the ground En = T ∞ − state. Due to the goodness of the n, � quantum numbers for Rydberg states, perturbations, lambda– doubling, and diagonal rotational constants and spin–orbit constants are all closely inter-related and eas­ ily understood. Since all Rydberg states belonging to the same series have similar potential curves, � �� �� � �2 Franck-Condon factors involving two Rydberg states are usually �v|v� � = δvv� , and 2µ v � r12 � v� = Be × � � 2 δvv� − 2x δ � + 6xr2 − . . . x = r − re where x is the vibrational displacement coordinate appropriate for re vv ±1 e evaluating vibrational matrix elements in the harmonic oscillator basis set. This means that all second order constants of the form � � �� � v, 2 Π |AL+ | v� , 2 Σ v, 2 Π |BL+ | v� , 2 Σ . αβ(ΠΣ) = EΠ − EΣ (v� ) v� simplify immediately to αβ(ΠΣ) = AΠ [(� − λ)(� + λ + 1)]1/2 BΠ [(� − λ)(� + λ + 1)]1/2 TΠ − TΣ 5.76 Lecture # 9 Supplement Page 8 Thus AΠ BΠ �(� + 1) TΠ − TΣ αβ(ΠΣ) = Similarly B2Π �(� + 1) . TΠ − TΣ This state of affairs is called “pure precession.” This occurs whenever n, � are good quantum numbers. Rydberg states and most electronic states of hydrides are well described by pure precession. For valence states, when n, � are not good quantum numbers, but are still good labels, a generalized pure precession result is obtained. Using the one electron product functions, all perturbation matrix elements and most second order sums may be accounted for by two constants β(ΠΣ)2 = a ≡ �n�λ|a�+ |n�λ − 1� b ≡ �n�λ|�+ |n�λ − 1� . The second order sums do not reduce as easily as for pure precession, because the potential curves do not always have the same shape and re . Sums over � v� �v|v� � EΠ (v) − EΣ (v� ) must be evaluated by computer. and � �v, Π|B|v� , Σ� EΠ (v) − EΣ (v� ) v�