(1/2) M α 2 α, ˆTe = i. 1 r i r j, ˆV NN = α>β

Transcription

1 Chemistry 26 Spectroscopy Week # The Born-Oppenheimer Approximation, H + 2. Born-Oppenheimer approximation As for atoms, all information about a molecule is contained in the wave function Ψ, which is the solution of the time-independent Schrödinger equation: ĤΨ( x, r ) = EΨ( x, r ) (4.) where x stands collectively for the spatial and spin coordinates of the n electrons in the molecule, and r denotes collectively the positions of all N nuclei in the molecule. In the non-relativistic limit, the total Hamiltonian for the molecule is Ĥ = ˆT N + ˆT e + ˆV Ne + ˆV ee + ˆV NN (4.2) ˆT N +Ĥel where ˆTN = α (/2) M α 2 α, ˆTe = i (/2) 2 i (4.3 4) ˆV Ne = α,i Z α R α r i, ˆV ee = i>j r i r j, ˆV NN = α>β Z α Z β R α R β (4.5 7) Atomic units have been used, in which h = m e = e =. ˆT N and ˆT e are the summed kinetic energy operators of the nuclei α with mass M α and the electrons i with mass m e, respectively, and ˆV Ne, ˆV ee and ˆV NN denote the summed Coulomb interaction energies between the nuclei and the electrons, between the electrons themselves, and between the nuclei themselves, respectively. Equation (4.) is a (3n+3N)- dimensional second order partial differential equation, which cannot be readily solved. Because the masses of the nuclei are much larger than that of the electrons, the nuclei move slowly compared with the electrons. It is usually (but not always!) a very good approximation to assume that the electronic energies (that is, the energies due to the motions of the electrons) can be determined accurately with the nuclei held fixed at each possible set of nuclear positions. In other words, it is assumed that the electrons adjust adiabatically to small or slow changes in the nuclear geometry. This approximation and its consequences were first examined by Born and Oppenheimer (927, Ann. Physik 85, 457), and has carried their names ever since. In this approximation, the total wave function is separable Ψ( x, R) = Ψ el ( x; R)Ψ nuc ( R) (4.8) into a nuclear part Ψ nuc that depends only upon the nuclear coordinates R, and an electronic part Ψ el that depends on the electronic coordinates x, but only parametrically on R. Ψ el is the solution of the electronic eigenvalue equation Ĥ el Ψ el ( x; R) = E el ( R)Ψ el ( x; R) (4.9) 5

2 where E el (R) is the potential energy surface, or, in the case of a diatomic molecule, the potential energy curve of the molecule in a particular electronic state. Substituting the Born-Oppenheimer wavefunction(7.8) into the Schrödinger equation, and using(7.9) gives: [ α where we have made the assumption that 2 α +E el ( ] R) E Ψ nuc ( R) = 0, (4.0) 2M α 2 αψ el Ψ nuc = Ψ el 2 αψ nuc +2 α Ψ el nuc α Ψ nuc +Ψ nuc 2 αψ el Ψ el 2 αψ nuc (4.) The neglect of the so-called non-adiabatic interactions α (the second and third terms in 4.) is usually justified, except when the electronic wave function changes rapidly with the nuclear coordinates. This can happen, for example, in those regions where two states interfere with one another (at a surface- or curve- crossing ), as we will see later in the class. Equation (4.0) is an eigenvalue equation for the nuclear motion, where E el (R) acts as the potential in which the nuclei move. We will illustrate the determination of the nuclear motion according to equation (4.0), which can only occur after the curves for E el (R) are obtained (or assumed), later this week. In the next few sections, we ll consider the simplest chemical bonds in the molecules H + 2 and H 2, in order to build up a quantitative picture of the chemical bond. 2. Calculation of electronic energies Asforatoms, themostgeneralmethodtosolveeq. (4.9)isthemethodofconfiguration interaction (CI), which is a straight-forward application of the (Ritz) linear variation method you learned in Ch 2a. The electronic wave function is expanded linearly as Ψ el ( x,... x n ) = M c k Φ k ( x,... x n ) (4.2) k= where Φ k are the expansion functions, loosely referred to as configurations, and c k the CI coefficients. If we substitute (4.2) into (4.9) and use the variational principle, which requires that the energy should be stationary with respect to variations in the wave function, we obtain the matrix eigenvalue equation Ĥ el c = E el c, (4.3) where (H el ) kl = < Φ k H el Φ l >, (4.4) and it has been assumed that the set of basis functions {Φ k } is orthonormal. If this set were complete, which implies k =,..., the calculated eigenvalues would be exact. In practice, of course, a truncated set {Φ k } is used, and the variational principle ensures that 6

3 the resulting (lowest) eigenvalue is always an upper bound to the exact energy. The lowest k roots of eq. (4.3) form the energies of the lowest k electronic states of the molecule. The functions Φ k are usually constructed as linear combinations of products of oneelectron functions u i ( x), where the combinations are chosen to satisfy the spatial and spin symmetry conditions and the Pauli principle. As for atoms, one can write the functions Φ k in terms of Slater determinants: Φ k ( x,... x n ) = AΞ k ( r,... r n )Θ k (s,...s n ) (4.5) where A is the antisymmetrization operator and Ξ k is a product of n spatial orbitals u i ( r): Ξ k ( r,... r n ) = u A ( r )u B ( r 2 ),...u X ( r n ). (4.6) Θ k is an n-electron eigenfunction of the total spin operators S 2 and S z, and is generally a linear combination of products of n one-electron spin functions α and β (see e.g. the case of two electrons discussed for the He atom in the Week # supplementary notes). The problem has thus been reduced to finding suitable one-electron functions u i ( r). Historically, two choices have been made. (i) Atomic orbitals (AO s), which are one-electron hydrogenic functions centered at the nuclei. This method has been come to be known as the Valence Bond (VB) method, and while useful suffers from the fact that the AO s are non-orthogonal. VB techniques are straightforward for diatomic molecules, but can become cumbersome for polyatomic species for which we will find that group theory offers considerable simplifications when molecular symmetry is considered. (ii) Molecular orbitals (MO s), which are delocalized functions over the whole molecule. The MO s are orthogonal, and are chosen such that one configuration Φ is (hopefully) already a good approximation to Ψ el ; that is, the goal is to choose the MO s such that expansion (4.2) becomes Ψ el (0.99) Φ + (0.0) Φ 2 + (4.7) How does one find the appropriate MO s? Starting from an ansatz (a educated guess) for the form of the configuration function Φ, the best MO s are expanded in a set of atomic orbitals m u i ( r) = φ µ ( r k )d µi, (4.8) µ= where the function φ µ is an atomic orbital centered at nucleus k. This is called the LCAO- MO (Linear Combination of Atomic Orbitals forming a MO) method. Note that, as for atoms, in the Hartree-Fock equations, the electrons are treated as independent particles. Although energies and properties of molecules can be computed at this level of theory, in most cases it is necessary to carry the calculation on to the higher level of CI, where the Hartree-Fock solutions are used as starting points. One of the problems in molecular structure is that most aspects of chemistry are only concerned 7

4 with small differences between large numbers. For example, the total binding energy of a molecule, or the excitation energy to the first excited electronic state, is usually less than 0.5% of the total energy. Let us consider two simple examples, the H + 2 here, and molecular hydrogen next. a) The H + 2 ion The simplest molecule, is the H + 2 ion, which consists of two protons held together by one electron. Thus, we do not have to deal with electron-electron repulsion, and the Hamiltonian is: H el = 2 /2 /r A /r B + /R (4.9) where r A = r R A, r B = r R B and R is the internuclear distance R A R B. e r r A B A R B Although this system can be solved exactly (within the Born-Oppenheimer approximation) in what are known as elliptical coordinates, it is more instructive to obtain an approximate solution using the variational method. If the electron is very close to A, the term /r B is small, and the Hamiltonian reduces to that for the hydrogen atom, since /R is fixed (as a parameter). So we expect: with φ A (r) = φ el (r) φ A (r) for r A << r B (4.20) π e r A (4.2) a hydrogen s function centered on nucleus A. Similarly, we expect φ el (r) φ B (r) for r B << r A (4.22) if the electron is close to nucleus B. Thus, we assume that the wave function in general can be written as a superposition of the hydrogen atomic wave functions: φ el (r) = c A ψ A (r) + c B ψ B (r). (4.23) This is the LCAO-MO approximation. Since the molecule has inversion symmetry, c A = ±c B, so that we find two orbitals: φ el ±(r) = c ± (ψ A ±ψ B ). (4.24) The + and combinations are called the bonding and antibonding orbitals, respectively. 8

5 A B A B φ + φ The normalization constant c ± can be found by integrating φ ± over the whole space: = φ el ± φ el ±d 3 r = 2±2S (4.25) c 2 ± where S(R) = ψ A ψ B d 3 r = (+R+R 2 /3)e R (4.26) is the overlap integral. The energies E el ±(R) can then be computed from: E el ±(R) = < φ el ± H el φ el ± > (4.27) = < φ A H el φ A > + < φ B H el φ B > ±2 < φ A H el φ B > 2±2S(R) All integrals in(4.27) can be computed exactly for the hydrogen wave functions as functions of the H H distance R, and the requisite mathematics are nicely outlined in McQuarrie, pp The first two terms are the same since the nuclei are both protons in this case, and so a factor of two can be divided out of both the numerator and denominator of eq. (4.27). The terms involving only one nucleus are often termed atomic orbitals, while the last term in the numerator is called an exchange (or two center) integral, which like the exchange integral encountered in the Hartree-Fock approach to atomic electronic structure is a quantum mechanical phenomena. We can then plot the resulting potential energy curves E± el as functions of R (see Fig. 4.). The lower electronic state E+ el is bound, in that the overall energy is lowered w.r.t. a ground state hydrogen atom and a proton; whereas the upper state E el has no minimum w.r.t. R and an energy larger than the separated atom + proton (and is thus unbound ). Qualitatively, the LCAO-MO wave function gives the correct result, but quantitatively, it is quite poor. Table 4. compares the computed and experimental values for the equilibrium internuclear distance R e and the dissociation energy D e. The wavefunction(4.23) can be improved by adding other atomic orbitals(for example 2s or 2p functions) to the expansion, and by varying the exponent of the hydrogenic functions (4.2). In analyzing why the latter approach is useful, it is helpful to consider what is known as the virial theorem, a topic we turn to next. 3. The Quantum Mechanical Virial Theorem & The Chemical Bond Classically, for a collection of particles, it was originally shown by Clausius (870) that N T = Fk r k k= 9.

6 el E E el - 0 R e R D e E el + Figure 4. The bound and unbound potential energy curves of the H + 2 ion. Table 4. Energetics of the H + 2 ion Property Calc. Expt./Exact Soln. R e (Å).3.03 D e (ev) where the brackets indicate an average over the ensemble and T is the kinetic energy and F k is the force on particle k located at position r k. For systems of particles that interact only via a central field potential, or V(r) = αr n, it can be shown that 2 T = n V, (4.28) an oft-cited and -used result. For a collection of harmonic oscillators, we see that < T >=< V >, while for self-gravitating gas or the hydrogen atom 2 < T >= < V >. How might we extend this result to quantum mechanics and to molecules? Here will we restrict ourselves to the Born-Oppenheimer approximation, and simply cite the fact that the eigenfunctions of the Hamiltonian are stationary states, for which it is possible to show that: d r i p i = 0 (4.29) dt i where the sum runs over only the electrons. This gives two terms, which after taking the expectation values and some manipulation yield 2 Ψ i h2 i Ψ 2m e 20 Ψ i V r i Ψ r i = 0. (4.30)

7 Well, we are getting closer since the first term in eq. (4.30) is just 2 < T e >, where the subscript denotes this is the kinetic energy of the electrons. Remember, in the Born- Oppenheimer approximation the nuclei are considered fixed, and we have three terms in the potential energy that must be considered (that is, V = V ee + V Ne + V NN ). Since V NN depends only on the nuclear coordinates, the derivatives of V NN w.r.t. the electron coordinates yields zero. The V ee terms depend only on the electron coordinates, and so is what is termed a homogeneous function. For such functions, and for V(r) = αr n, it can be shown that Ψ i r i V ee r i Ψ = n < V ee > = < V ee > (4.3) for the Coulomb interaction. The messy terms involve V Ne, and the manipulations involves things like Euler s theorem and a number of other steps. Here, we ll just cite the result which is: Ψ i r i V Ne r i Ψ = n < V Ne > + α R α Ψ α V Ne Ψ. (4.32) The V NN terms do depend on the nuclear coordinates R α, and in getting to a final expression that is conveniently in terms of the total energy it is helpful to use the Hellman- Feynman theorem, which states which can be used to find that E E p = Ψ Ĥ p Ψ, (4.33) R α = Ψ α (V Ne +V NN ) Ψ. (4.34) Putting all this together, we find that within the Born-Oppenheimer approximation the virial theorem can be cast as 2 T e + V + α R α E R α = 0. (4.35) The last term in eq. (4.35) is clearly related to the force on the nuclei, which only equate to zero for stationary points on the potential energy surface (minima, maxima, saddle points); and so these are only only locations for which the idealized virial theorem relationship of < V >= 2 < T > is obtained. The virial theorem can be used to test the quality of wavefunctions, and in this regard how does the simple function outlined in (4.23) do for the H + 2 ion? Not all that well, as it turns out! For simple atomic hydrogen wavefunctions without modification, as McQuarrie 2

8 shows on pp the ratio of the potential to kinetic energy is -.6 and not -2 as it should be, at the theoretical potential energy curve minimum of.3 angstrom. If a s Slater orbital is used instead, namely φ = ζ 3 /πe ζr, then we can optimize both the wavefunction coefficients (which must be related by symmetry as above) and exponents to optimize the energy. As outlined in McQuarrie, pp , the value of ζ which produces < V > / < T >= 2.0 is.238. At this value, the predicted internuclear distance becomes.06 Å (down from.3 Å, and in much better agreement with the exact value of.03 Å) and the energy minimum is in better agreement with the experimental value as well (see Fig. 0.7). Matching both the energy value and bond length requires configuration interaction, about which we ll have more to say next time. What does all this tell us about chemical bonds? As McQaurrie discusses, the answer is that the variation of both the kinetic energy and potential energy w.r.t. the internuclear distance is critical. Figure 0.8 in McQuarrie shows the results for H + 2, Figure 4.2 shows the energetics for H 2, which we ll tackle next. Figure 4.2 Virial theorem plots of the kinetic, potential, and total energy for H 2. As the nuclei approach each other, the potential energy rises at first, it is the kinetic energy that provides the long range attractive force. Near the equilibrium bond length the potential energy dominates, but it is the balance of the K.E. increase and the decrease in V that sets the energy minimum. 22

9 04-08Apr206 Chemistry 2b Spectroscopy Week # MO Theory for H 2, Electronic Nomenclature for Molecules. The Chemical Bond & Molecular Hydrogen Recall that the polyatomic Hamiltonian is: Ĥ = ˆT N + ˆT e + ˆV Ne + ˆV ee + ˆV NN (5.) ˆT N +Ĥel where ˆTN = α (/2) M α 2 α, ˆTe = i (/2) 2 i (5.2 3) ˆV Ne = α,i Z α R α r i, ˆV ee = i>j r i r j, ˆV NN = α>β Z α Z β R α R β (5.4 6) Atomic units have been used, in which h = m e = e =. The simplest molecule in which electron-electron repulsion (and thus electron-electron correlation) is present is H 2. The electronic Schrödinger equation for even this simplest of multi-electron molecules cannot be solved analytically. As a first approximation, it is typical to follow the Hartree-Fock approach and use Slater determinants to create what are now properly anti-symmetrized molecular orbitals φ el ± for the H + 2 molecule as a starting point, and form an H 2 wavefunction from these to be variationally optimized. In the ground state of H 2, both electrons will be in the energetically more favorable orbital φ el +. Because of the Pauli principle, the two electrons must then be paired as a singlet, which for this simple two electron case we can write as a product of a spatial wavefunction and a spin wavefunction, just as for the He atom: Φ = 2 φ +()φ + (2){α β 2 β α 2 }, (5.7) withφ + givenby(4.24). ThiswavefunctionworkswellnearR e, butasmcquarriedescribes it fails terribly at large R, where it partly represents an H and an H + ion pair rather than two ground-state H atoms. This can be seen by expanding (5.7) into the atomic orbitals φ A and φ B : Φ = (φ A +φ B ) (2+2S) /2 (φ A +φ B ) 2 (2+2S) /2 = 2+2S {φ Aφ A +φ A φ B +φ B φ A +φ B φ B } = 2+2S {(φ Bφ A +φ A φ B )+(φ A φ A +φ B φ B )}. (5.8) The first two terms correspond to one electron/h nucleus, which is appropriate, but the second pair of terms correspond to both electrons being on one nucleus or another (Why are two terms appropriate for each of these limits?). Recall that the ionization energy of a hydrogen atom is 3.6 ev, and that it s electron affinity is 0.75 ev, and so an equal 23

10 contribution of these so-called atomic (or valance bond) and ionic components of the simplest molecular orbital wavefunction has an asymptotic limit for large R that is in error by considerably more than a chemical bond! This problem can be remedied by adding a second configuration made up of the φ combination: Φ 2 = 2 φ ()φ (2){α β 2 β α 2 } (5.9) so that the CI wavefunction becomes Ψ el g = c Φ + c 2 Φ 2 (5.0) andthecoefficientsc i canbeobtainedbydiagonalizingthe2 2interaction(secular)matrix (for each/all separations R). There are actually six possible spatial-spin wavefunction combinations, but as McQuarrie nicely outlines (pp ) most terms are zero by symmetry, leading to eq. (5.0). As R, c, c 2 / 2, as they should, and near the equilibrium bond length c c 2, in accordance with chemical intuition. As for H + 2, this level of treatment gives a correct qualitative description of the ground state of H 2, with a binding energy 3. ev and a bond length of 0.76 Å (compared to experimental values of 4.74 ev and 0.87 Å). Thus, the simplest level of theory for H 2 does not work nearly as well as it does for atoms. To obtain good quantitative agreement, it is necessary to include many more functions in the LCAO-MO function (4.7). As we will see below, the individual molecular orbitals in diatomic molecules are characterized by the component of the electronic orbital angular momentum along the chemical bond. The relative contributions of the kinetic and potential energy to the attractive, singlet potential energy curve for molecular hydrogen are shown in Figure 4.2. One can also construct a triplet H 2 state by putting one electron in the φ + orbital and one in the φ orbital (thus combining the symmetric spin wavefunction with an anti-symmetric spatial wavefunction). The resulting potential curve in this case is repulsive or unbound, that is, it has the same general shape as the E el potential of H + 2. Note that these potential curves are the same for the isotopes HD, D 2, etc... under the Born-Oppenheimer approximation, and that there are no states just above the dissociation energy for R R e. That is, the energy of the repulsive state is lifted above the separated atom limit by an energy comparable to the chemical bond. This has important implications for the means by which molecules interaction with radiation, a topic we ll turn to later in the course. el φ - Energy φ el + Φ Φ 2 Ψg 3 Ψ (triplet state) (singlet ground state) Note also that for atoms, it was argued that electrons with aligned spins (large total spin) lie lowest in energy (Hund s rule ). Now we find for H 2 that it is the singlet state 24

11 that binds, while the triplet state does not. This paradox is explained by the fact that for molecules, it is the electron density between the nuclei that leads to binding, and this effect outweighs the lower e e repulsion energy in the high spin states. Another point of interest involves the large R behavior of the internuclear potential. In a second-order perturbation expansion, it is found that two H-atoms attract each other with a R 6 van der Waals potential. Thus, even the triplet state eventually becomes attractive at very large R, but with very shallow well depth that results in no bound states although weak bonds are found in other molecules. Note also that not all pairs of atoms can form bound molecular states; for example, two ground state He atoms cannot form a bound electronic state. 2. Nomenclature for Electronic States of Molecules Recall that for atoms, we let L i and S i be the individual electron angular momenta and spin. Given the spherical symmetry of atoms, the total angular momentum and spin) operators, L = n i L i and S = n i S i, commute with Ĥ, so L and S are good quantum numbers. We indicate the total orbital angular momentum by a letter as follows: L S P D F G H I K L... The individual orbital occupancy is given in small letters, i.e. a s 2 2s 2 2p3d configuration gives rise to P, D, and F states. The quantity 2S +, the total spin degeneracy (multiplicity), is written as a left superscript to the letter designating L. For the s 2 2s 2 2p3d configuration, the two spins may be parallel or anti-parallel, which gives rise to the states: P, 3 P, D, 3 D, F, 3 F Atomic states arising from the same electron configuration and having the same value of L and of S are said to belong to the same term. States belonging to different terms have different energies because of electron-electron repulsion, and the lowest state may be found according to Hund s Rules, which state that: The term with the highest spin multiplicity is lowest in energy; if there is more than one term with the highest multiplicity, then the term with the highest multiplicity and largest value of L lies lowest. Remember, Hund s rule works only for the lowest state (for example, the experimental ordering of the s 2 2s 2p 3 configuration of C is 5 S< 3 D< 3 P< D< 3 S< P!!). The physical basis of Hund s rule is that high spin states correlate with anti-symmetric spatial wavefunctions that, on average, have average electron-electron distances which are larger (and hence smaller electron-electron repulsion). This, of course, is due to the Pauli principal and the fermion nature of electrons. The total electronic angular momentum J of the atom is the vector sum of L and S, or J = L+S, so J = L+S, L+S,..., L S. (5.) The value of J is written as a right subscript on the term symbol. Thus for a 3 P term, L+ and S =, and we get 3 P 0, 3 P, and 3 P 2 levels. In the above Hamiltonian, levels of the same term have the same energy. The spin-orbit interaction which splits the levels, and gives rise to the so-called L S, or Russell-Sanders, and j j coupling limits (where the 25

12 spin-orbit energy is, respectively, much, much smaller than or comparable to the electronelectron repulsion). For most heavy atoms, the situation is intermediate between L S and j j coupling. Similar coupling schemes are used in molecular spectra, which we ll look at next for diatomic molecules. Nomenclature for Molecular Electronic States For historical reasons, electronic states are often given letter designations to label them roughly in order of energy or discovery. For diatomics, the ground state is denoted with the letter X. The letters A, B, C, D, are reserved for the lowest excited electronic states of the same spin multiplicity as X, usually in order of increasing energy. The letters a, b, c, d,.. denote the lowest excited states of different spin multiplicity from the ground state. For polyatomic molecules the conversion is the same, except that all letters have a tilde superposed; e.g., X, Ã, b,... because the un-accented letters are needed to add symmetry labels to the electronic states as we ll see when we cover group theory. As we outlined above for H 2, it is the singlet states which tend to create bonding orbitals while the triplet states generate repulsive curves, for closed shell molecules. So, electron-electron repulsion is not the most important aspect here. Instead of Hund s rule and L S or j j coupling, for molecules the addition of angular momenta are governed by what are termed Hund s coupling cases, which we review next. Hund s Coupling Cases Unlike atoms, diatomic molecules have cylindrical symmetry, which means that L is no longer a good quantum number; only the component Λ of the orbital angular momentum along the internuclear axis is defined, where Λ can be 0,, 2, 3,... For orbitals that are filled, they make no contributions to Λ since the electrons are paired (just as filled atomic shells do not need to be considered when constructing term symbols). Because the component of the orbital angular momentum of the electrons and the overall Hamiltonian commute, the wavefunctions, and thus the diatomic molecular orbitals, can be characterized according to their value of Λ (called m, pp McQuarrie). By analogy with the atoms, lower case Greek letters are used for the S, P, D, F orbitals to give MOs with Λ=0,, 2, being given the designation σ, π, δ... Also by analogy with atoms, Λ is the vector sum of the angular momenta λ i of the individual electrons in the molecule, or Λ = i λ i. All electronic states with Λ > 0 are doubly degenerate. Classically, these degeneracy can be though of as being due to electrons orbiting clockwise or anti-clockwise around the internuclear axis, the energy being the same in both cases. If Λ = 0, there is no orbiting motion, and no degeneracy. As for atoms, the states are designated by their value of Λ using the Greek equivalents of S, P, D, or Λ Σ Π Φ... L Λ = M L A Λ B z 26

13 OnealsodistinguishesbetweenΣ + andσ states,dependingonwhetherthemolecular orbital is symmetric (+) or anti-symmetric ( ) with respect to reflection across any plane containing the internuclear axis. For Λ > 0, one will have +, the other, but the symbolism Π ±, ± is not often used. S Λ = 0 S A Λ n.e. 0 B Σ z If the diatomic molecule has two identical nuclei (H 2, C 2,..), subscripts u and g must be added to distinguish functions that are gerade or ungerade upon inversion through thecenterofthemolecule. Justasforatoms, thetotalspins oftheelectronsinaparticular electronic state is indicated by the multiplicity 2S + as a superscript. Thus, the ground state of H 2 is Σ + g, with two electrons in the σ g s 2 M.O. Unfortunately, the projection of S along the internuclear axis is also called Σ. For Λ = 0, Σ is not defined, that is there are no torques on S, and so it just sits there, as is shown in the figure above. For Λ 0, Σ = S,S,..., S +, S, and the internal magnetic field set up causes S to precess, coupling the orbital and spin momentum (a la Ĥs.o. in atoms). The total angular momentum is called Ω, and Ω = Λ + Σ, (5.2) and is placed as a subscript as J is for atoms. For example, a state might look like: 3 3 Λ = 2 Ω = 3 Σ = A > 0 regular A < 0 inverted 3 2 Λ = 2 E A Ω = 2 Σ = 0 E o 3 3 Λ = 2 Ω = Σ = E = E + A Λ Σ o where the splitting between the Ω sub-states arises from the spin-orbit interaction. All of the interactions we have just considered are without nuclear rotation or nuclear spin, topics we ll turn to next time. 27