Motion equations nuclear

The concept of relaxation time was introduced to the vocabulary of NMR in 1946 by Bloch in his famous equations of motion for nuclear magnetization vector M [1] ... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

NMR spectroscopy provides spin-lattice (ri) and spin-spin (Tj) relaxation times. Making appropriate assumptions with regard to the magnetic interactions responsible for the relaxation process, these relaxation times can be related to molecular motions. Since nuclear spin relaxation results from all processes which cause a fluctuation in the magnetic field at the nucleus, the correlation function will generally correspond to more than one kind of motion involving all possible interactions. The equations for the relaxation times are generally of the form... 

The exact Schrodinger equation of motion, equation (1), may be equivalently Stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Yj(x, R) is expanded in terms of the complete orthonormal set of functions y>i(x R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the x (R)6... 

Combined with appropriate asymptotic conditions, these are the nuclear motion equations which must be solved when two electronically adiabatic PESs are involved in the process. As for the one-state approximation, the existence of a conical intersection between the eftq) and Ejd(q) PESs requires that additional restrictions be imposed on x d and Xjd-These will be discussed in Sec. III.B. 

Under the conditions of validity of the two-electronically-adiabatic-state approximation it is possible to change from the i]/al,ad(r q) (n = i, j) electronically adiabatic representation to a diabatic one 1,ad(r q) (n = i, j) for which the VR Xn(R) terms in the corresponding diabatic nuclear motion equations are significantly smaller than in the adiabatic equation or, for favorable conditions, vanish [24-26]. Such an electronically diabatic representation is usually more convenient for scattering calculations involving two electronically adiabatic PESs, but not for those involving a single adiabatic PES. This matter will be further discussed in Sec. III.B.3 for the case in which a conical intersection between the E ad(q) and Ejad(q) PESs occurs. 

What is the significance of (14.76) In the Born-Oppenheimer approximation, U xa, y , Za,. ..) is the potential-energy function for nuclear motion, the nuclear Schrodinger equation being... 

Inserting this form of the total wavefunction into the time-dependent Schrodinger equation and using the above orthonormal property, one obtains the standard coupled equations of motion for nuclear wavefunctions. .d... 

With the above Hamiltonian, Eq. (6.10), an analogy to purely classical mechanics brings about the canonical equations of motion for nuclear classical variables (R, P) as [492]... 

With this energy as potential energy for the nuclear motion the nuclear Schrodinger equation in the Born-Oppenheimer approximation, Eq. (2.12), becomes... 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Here, the first factor (r, R) in the sum is one of the solutions of the electronic BO equation and its partner in the sum, Xt(R) is the solution of the following equation for the nuclear motion, with total eigenvalue... 

---