- Principles of the Quantum Control of Molecular Processes. - Term Values of the Vibrating Rotor. - Matrix Elements in the Born-Oppenheimer Approximation Structure of the Spectra of Diatomic Molecules. - of the lowest Bohr orbit in the hydrogen atom. - averaged over the motion of the electrons. - (2.26) is the matrix element of the perturbation operator T. - E t (R) of the unperturbed states. - 2.2 An example for the breakdown of the Born- Oppen heirner approximation.. - A w above the minimum of the potential curve. - Exact Treatment of the Rigid H i Molecule. - 2.7 Elliptic coordinates of the H: molecular ion.. - e of the electron. - For the projection quantum number R of the total angular momentum we obtain. - b) A s quantum number C = Ms of the total spin projection Msri 3. - a suitable average of the sums Ci bij, l i x s j for j = 1. - mentum of the same electron, and. - 0 a well-defined state of the united atom. - Electron configuration of the BH molecule. - Usually, these are the ground states of the two atoms. - (2.13a), of the Schrodinger equation (2.4) for fixed nuclei,. - The magnitude of the splitting for 4. - Depending on the relative phase of the two oscillations. - Deficiencies of the Simple LCAO Method. - of the kinetic energy T ( R. - 2.29 Optimization of the contraction parameter v ( R. - From a comparison of the E ( R ) curves in Fig. - (2.3), rendering the separation of the many-electron wavefunction. - The Hamiltonian of the H2 molecule (Fig. - The expectation value of the total energy is then. - (2.90) can be solved and written as a function of the. - As in the case of the H: ion, the calculated values for the bond energy D. - We can correct for the overestimation of the ionic contribution a( l)b( 1. - of the i th electron for all electrons. - for the movement of the nuclei in the potential E:(R) of the electronic state (.I. - (3.7) the energies of the rigid rotor,. - E(B) of the separated atoms A and B and the electronic energy E(AB) of the molecule at the minimum of the potential curve.. - The repulsive part of the potential. - (3.22), we obtain for the energies E(w) of the vibrational levels w. - x exp( -c2 /2) of the harmonic oscillator. - As I$vib(R) l2 dR is the probability of finding the nuclei at an internuclear distance between R and R + dR, the mean value (quantum-mechanical expectation value) of the internuclear distance is. - of the potential expansion which was determined by Dunham [3.9].. - If the physical meaning of the coefficients in Eq. - M I +M2) of the molecule. - (3.52), the mass dependence of the Dunham coefficients. - Expansion of the integrand yields. - Introduction of the de Broglie wavelength. - E( v,J)/hc in the form of the Dunham expansion. - Note: As the expansion of the potential Eq. - 3.13 Explanation of the RKR procedure. - The bond energy of the molecule is then (neglecting zero-point energy) b2. - Ik) of the corresponding molecule.. - of the intensity I ( z ) with increasing absorption path length z, where a ( v ) is the frequency-dependent absorption coefficient. - w , k depend on the spectral energy density p( v) of the radiation field.. - of the nuclei to the dipole moment. - of the nuclear framework.. - 4.2 Structure of the Spectra of Diatomic Molecules 129 I. - Structure of the Spectra of Diatomic Molecules. - 4.3 Orientation of the molecular axis (Z axis) in the labora- tory frame X , Y , Z. - of the nuclear frame- work,. - 4.2 Structure of the Spectra of Diatomic Molecules I 131. - of the vibrational function ?,bVib(R. - the vibrational part of the matrix element Eq. - 4.2 Structure of the Spectra of Diatomic Molecules I 133. - Hence, we obtain for the transition probability of the whole transition J. - 4.2 Structure of the Spectra of Diatomic Molecules 135 I. - 4.2 Structure of the Spectra of Diatomic Molecules 137 I. - The wavenumbers of the rotational lines are. - 4.6 Vibration-rotation spectrum of the (v. - The electronic part of the matrix element,. - 4.2 Structure of the Spectra of Diatomic Molecules I 139. - The square modulus of the first integral,. - 4.2 Structure of the Spectra of Diatomic Molecules I 141. - of the difference Hamiltonian H. - The uncertainty width AR of the R centroid. - 4.2 Structure of the Spectra of Diatomic Molecules I 143. - comparison of the R centroid approximation with exact values. - 82) in the DIEu state of the CSZ molecule for the vibrational transitions IZ,,(w. - 4.2 Structure of the Spectra of Diatomic Molecules I 145. - 4.2 Structure of the Spectra of Diatomic Molecules I 147. - The results of the calculations for the line intensities S R ( J. - 4.2 Structure of the Spectra of Diatomic Molecules I 149. - 12,14) of the D ‘n. - I ) maxima of the vibrational wavefunction. - it is called the natural linewidrh of the transition.. - The natural linewidth of the transition is then Au. - (4.94b), for the line profile of the molecule instead of Eq. - These microscopic contributions of the individual molecules to. - J is always the quantum number of the lower level.. - The term values of the involved levels I J. - where vo is the wavenumber of the exciting transition. - is the total density of the molecules and Z = C e &. - The ratio of the two statistical weights is therefore. - 5.1 shows all symmetry operations of the H20 molecule. - 5.1 Symmetry operations of the H 2 0 molecule.. - 5.3 Some symmetry elements of the benzene molecule C6H6.. - 5.1 Multiplication table of the group GV.. - Remark: For the nuclear framework of the H 2 molecule, the operation. - 5.2 Multiplication table of the group C3v.. - 5.4 Multiplication table of the group C2h.. - b) SF6 as an example of the group oh.