Molecular Spectra and Molecular Structure - Lec 11S

Lecture # 11 Supplement Contents A. Magnetic and Electric Effects . . . . . . . . . . . . . . . . . . . . . . . . 1 B. Related Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2. A. “Reflection and Interference Structure in Diatomic Franck-Condon Distri­ butions” by Joel Tellinghuisen . . . . . . . . . . . . . . . . . . . . . . . . 7 “The Calculation of Rotational Energy Levels and Rotational Line Inten­ sities in Diatomic Molecules,” by Jon T. Hougen . . . . . . . . . . . . . . 8 Magnetic and Electric Effects Stark & Zeeman Effects, Magnetic & Electric Dipole Transitions Static magnetic and electric fields alter the energies of |ΩJM ; ΛSΣ� basis functions. Oscillating electric or magnetic fields (photons, coherent and incoherent radiation fields) cause transitions between basis functions. The first step in describing magnetic and electric effects is to write a Hamiltonian operator. The energy of a dipole in a static field is µ · E ). This operator is (or µ · H ∼ ∼ ∼ ∼ more complex than any we have dealt with so far because it is a vector product of two operators, one of which is necessarily referred to laboratory fixed coordinates and the other to molecule fixed coordinates. An electric field interacts with a molecular electric dipole moment; a magnetic field with a molecular magnetic dipole moment. The electric dipole is associated with the distribution of electronic charge along the internuclear axis. In a diatomic molecule the centers of gravity of positive and negative charge must both be on the internuclear axis; when these centers do not coincide an electric dipole moment exists. The magnetic dipole is associated with circulating electronic charge and S (orbital and spin angular momenta, L ). ∼ ∼ 1 Page 2 The energy of a dipole in a static field (or the instantaneous interaction of a dipole with an oscillating field) is: or µm · H. (1) µe · E ∼ ∼ ∼ ∼ A static field may be chosen to be Z directed (lab coordinates). E = Ek̂ ∼ µe · E = µ Z E ∼ H = Hk̂ ∼ µm · H = µmZ H ∼ (2) (3) where µZ and µmZ are the lab fixed Z components. An oscillating field may be chosen to be Z directed (Z polarized) or it can be a complicated superposition of X, Y , and Z directed fields. = µX EX + µY EY + µZ EZ . µe · E ∼ ∼ (4) The relation between the lab fixed X, Y, Z component of µ and the molecular x, y, z directions (internuclear axis is z) is given by direction cosine matrix elements. µZ = µz cos(Z, z) = µz cos(Z, J) cos(J, Jz ) (5) Even though it is difficult to imagine components of a dipole moment of a diatomic molecule in directions perpendicular to the internuclear axis, such components exist due to rotational distortion from inherent cylindrical symmetry about the z axis — the J vector defines a new direction. Thus perpendicular components of µ were erroneously omitted from equation (5). µZ = µz cos(Z, J) cos(J, Jz ) + µx cos(Z, J) cos(J, Jx ) + µy cos(Z, J) cos(J, Jy ) (5a) Convert to ± molecular components for convenience: µ± = µx ± iµy αZ± = cos(Z, J)[cos(J, Jx ) ± i cos(J, Jy )]. (6) (7) Thus 1 (8) µZ = µz αZz + [µ+ αZ− + µ− αZ+ ] 2 The direction cosines, α, operate only on the rotational part of the wave function, |ΩJ M �, while the µi operate only on the electronic part, |nΛSΣ� (and to the extent that µ is a function of R, on |v�). We will list the matrix elements and transformation properties (with respect to σv , C2 , and i) of the µ’s and α’s. See Table 6, page 31 of Hougen for all of the matrix elements of the direction cosines α. The matrix elements of α diagonal in MJ are needed for Stark and Zeeman effects and are listed in Table 1 on the next page. (The matrix elements of αZz and αZ± are listed, see Hougen for αXz , αX± , αY z , αY ± ). ΩM J(J+1) M [(J �Ω)(J ±Ω+1)]1/2 J(J+1) [(J+Ω+1)(J −Ω+1)(J+M +1)(J −M +1)]1/2 (J+1)[(2J+1)(2J+3)]1/2 �[(J ±Ω+1)(J ±Ω+2)(J+M +1)(J −M +1)]1/2 (J+1)[(2J+1)(2J+3)]1/2 �J � Ω ± 1M |αZ ± |JΩM � Table I J� = J �J � ΩM |αZz |JΩM � J� = J + 1 ±[(J �Ω)(J �Ω−1)(J+M )(J −M )]1/2 J[(2J+1)(2J −1)]1/2 [(J+Ω)(J −Ω)(J+M )(J −M )]1/2 J[(2J+1)(2J −1)]1/2 J� = J − 1 Page 3 Page 4 The transformation properties of the direction cosines are: σv αZz = −αZz (9a) σv αZ± = −αZ+− (9b) C2 αZz = −αZz (10a) C2 αZ± = −αZ+− (10b) Since i = C2 σv , the direction cosines must be invariant under i. At this point the treatments for electric and magnetic fields diverge due to the different forms of r e (the electron coordinate) but the magnetic the dipole operators. The electric dipole operator is ∼ dipole operator is µ0 (L + 2.0023S) where µ0 is a constant (the Bohr magneton): ∼ ∼ µ0 = 1.39967 MHz/Gauss. (11) The electric dipole moment is given in units of Debyes. 1D = 10−18 esu·cm= 3.33564 × 10−30 Coulomb·meter. A positive and negative charge separated by 1 Å corresponds to a dipole moment of ∼5D. The integral over electron coordinates: � ∗ (12) µz = φ� ze φdz is always an unknown (variable) parameter, but the integral over electronic (and spin) coordinates corresponding to the magnetic dipole moment contains only angular momentum operators. Thus the magnetic dipole moment can always be related to familiar matrix elements of L and S between ∼ ∼ |nΛSΣ� basis functions. Electric Dipole Matrix Element and Transformation Properties. Note that µ operates on electron coordinates only, not on spin, thus it can have no matrix elements off-diagonal in S or Σ. �n� Λ� S � Σ� |µz |nΛSΣ� = δSS � δΣΣ� δΛΛ� µz (S, Σ, Λ) �n� Λ� S � Σ� |µx ± iµy |nΛSΣ� = δSS � δΣΣ� δΛ� Λ±1 µ± (S, Σ, Λ) (13) (14) σv µz = µz (15a) σv µ± = µ� (15b) C2 µz = −µz (16a) C2 µ± = −µ� (16b) iµz = −µz (17a) iµ± = −µ± (17b) Page 5 Magnetic Dipole Matrix Elements and Transformation Properties. The magnetic dipole operator should be regarded as the sum of two operators µ0 L + 2.0023µ0 S. ∼ ∼ µ0 �n� Λ� S � Σ� |Lz |nΛSΣ� = δSS � δΛΛ� δΣΣ� µ0 Λ (18a) µ0 �n� Λ� S � Σ� |L± |nΛSΣ� = δSS � δΣΣ� δΛ� Λ±1 µ0 L+ (S, Σ, Λ) (18b) 2.0023µ0 �n� Λ� S � Σ� |Sz |nΛSΣ� = δSS � δΣΣ� δΛΛ� 2.0023µ0 Σ (19a) 2.0023µ0 �n� Λ� S � Σ� |S± |nΛSΣ� = δSS � δΣ� Σ±1 δΛΛ� 2.0023µ0 [(S � Σ)(S ± Σ + 1)]1/2 (19b) Note that matrix elements off-diagonal in Λ and Σ by ±1 exist, but not off-diagonal in both Λ and Σ. No matrix elements are off-diagonal in S. Recall that there are no matrix elements of the electric dipole moment off-diagonal in Σ. In molecules which are close to the case (b) limit (Λ = 0 or small spin-orbit coupling) S can be regarded as quantized with respect to lab coordinates rather than molecular coordinates (after all, if Λ = 0 and/or AL·S is small, S has no way of knowing where the internuclear axis is). If we could ∼ obtain direct matrix elements of µ0 SZ rather than µ0 αZz Sz + µ0 αZ± S� it would simplify matters. The representation where S is lab quantized, that is where ∼ SZ |SSZ � = MS |SSZ � (20) 2 is the representation where the matrix of (J∼ − S )2 = N is in diagonal form. Thus, if the transfor­ ∼ ∼ mation T diagonalizes (J∼ − S )2 ∼ � 2� 2 T= N δ T† (J∼ − S) ∼ ∼ ij ij (21) then the transformation of the SZ matrix, evaluated in the |N JSSz � basis will return us to the case (a) representation with which we are familiar: TSZb T† = SZa . The SZb matrix is diagonal, with dimension 2S + 1 and values S, S − 1, · · · − |S|. (22) Page 6 The transformation properties of the magnetic dipole operator are σv µz = µ0 σv (Lz + 2.0023Sz ) = −µz (23a) σv µ± = µ0 σv (L± + 2.0023S± ) = −µ� (23b) iµz = µ0 i(Lz + 2.0023Sz ) = µz (24a) iµ± = µ0 i(L± + 2.0023S± ) = µ± (24b) C2 µz = µ0 C2 (Lz + 2.0023Sz ) = −µz (25a) C2 µ± = µ0 C2 (L± + 2.0023S± ) = −µ� (25b) The transformation properties of σv , C2 , i on (αµ) for electric and magnetic dipole operators are summarized below: TABLE 2 Electric Magnetic −αZz µz αZz µz σv (αZz µz ) σv (αZ± µ� ) −αZ� µ± αZ� µ± i(αZz µz ) i(αZ± µ� ) −αZz µz −αZ± µ� αZz µz αZ± µ� C2 (αZz µz ) C2 (αZ± µ� ) αZz µz αZ� µ± αZz µz αZ� µ± Thus nonzero electric dipole matrix elements are (total parity) (1) + ↔ − (2) g ↔ u (3) s ↔ s , a ↔ a. Additional selection rules are due to dependence of µelectric on electron orbital (not spin) coor­ dinates (4) (5) Σ + ↔ Σ+ ΔΣ = 0 , , Σ − ↔ Σ− ΔS = 0. The nonzero magnetic dipole matrix elements are (1) + ↔ + (2) g ↔ g (3) s ↔ s , − ↔ − (total parity) , u↔u , a ↔ a. Considering the magnetic dipole as two additive terms, additional selection rules are derived. � � The orbital part of µ allows using σvΛ Page 7 (4a) ΔS = 0 Σ+ ↔ Σ− . , ΔS = 0 , and whenever ΔΛ = 0 and Λ = 0, then � � The spin part of µ allows using σvΣ (4B) (5) ΔS = 0, ΔΣ = 0, ±1, ΔΛ = 0 whenever ΔΣ = ±1 and Λ = 0, Σ+ ↔ Σ+ , Σ− ↔ Σ− but for ΔΣ = 0 and Λ = 0, Σ+ ↔ Σ− . Stark and Zeeman effects are much more complicated than the zero-field problem because of the large number of non-zero matrix elements. Since ΔJ = ±1 matrix elements are non-zero, matrices will be infinite and a truncation at some value of J is necessary, or a simplification of the Hamiltonian matrix is made treating only nearby interacting zero-field basis functions. The approach to each problem is different and no general method exists. B. 1. Related Papers “Reflection and Interference Structure in Diatomic Franck-Condon Distributions” by Joel Tellinghuisen ABSTRACT J. Mol. Spect. 103, 455-465 (1984). The Franck-Condon distributions for diatomic radiative transitions from a single vibrational level of a given electronic state to all possible levels (bound and free) of a second electronic state exhibit either “reflection” or “interference” structure. In reflection structure there is a one-to-one mapping of peaks in the initial state probability distribution into peaks in the spectrum. No such simple relationship is known for interference structure, originally termed “internal diffraction” by Condon [Phys. Rev. 32, 858-872 (1928)]. A semiclassical treatment of the quantum mechanical overlap integrals shows that the condition for reflection structure is a monotonic difference potential in the range of internuclear distance sampled by the initial wavefunction, whereas interference structure occurs when a “polytonic” difference potential is sampled. The qualitative validity of the semiclas­ sical treatment is illustrated through quantum calculations which show that the Franck-Condon integrals accumulate near the points of stationary phase. Page 8 2. “The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules,” by Jon T. Hougen ABSTRACT National Bureau of Standards Monograph 115 52 pp. (1970). This monograph describes procedures for making quantum mechanical calculations of rotational energy levels and rotational line intensities in diatomic molecules. The procedures are illustrated by sample calculations. A familiarity with the material of this monograph should enable a practicing electronic spectroscopist to carry out, though in a rather mechanical way, his own theoretical calculations for molecules under experimental investigation. A true understanding of the procedures described below can only be acquired by studying the theoretical reference material cited. These references are not exhaustive; they represent sources which the author finds convenient and instructive. Unfortunately, the material in the references is sometimes presented in a notation different from that used here. In addition, it is sometimes slightly too general or slightly too specific to apply directly to diatomic molecules. The material of this monograph is aimed at electronic spectroscopists who have had the equivalent of one semester of graduate-level quantum mechanics. Finally, I should like to point out, that the procedures described here for carrying out calculations are not new. Neither do they represent all possible correct procedures. They do represent, however, a unified approach to the problem, which, in the opinion of the author, is easier to understand than the original literature cited, and less likely to lead to error than some of the alternative calculation procedures. Note added in proof: The reader is referred to the book Rotational Structure in the Spectra of Diatomic Molecules (Adam Hilger Ltd., London, 1969) by I. Kovács, which just appeared and which represents another author’s discussion of much of the material in this monograph.