Quantum molecular functionals

An analogous role has been played by other scientists in strengthening the ties between quantum chemistry of type I (and type II) with the area corresponding to biochemistry (or complex molecular systems in general), a task made more difficult by the explosive growth of structural and functional information about biomolecular systems. It is worth to remark here that such a fruitful use of quantum chemical concepts in biology has requested the extension of the methods to approaches different from quantum molecular theory in the strict sense introduced before. We shall comeA back to this remark later. 

Sola, M., J. Mestres, R. Carbo, and M. Duran. 1996. A Comparative Analysis by Means of Quantum Molecular Similarity Measures of Density Distributions Derived from Conventional ab initio and Density Functional Methods. J. Chem. Phys. 104, 636. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... 

We have seen that the ab initio self-consistent quantum mechanical functional methods such as DFT/B3LYP with the chosen 6-31+G(d,p) basis sets are well suited to calculate reasonable molecular ion structures and vibrational spectra of these ions. The results obtained by us or others have indicated that the neglect of the presence of cation-anion interactions is a reasonable approximation for a rather successful prediction of the Raman spectra. Based on such calculations, detailed and reliable assignments of the spectra can be given and information on conformational equilibria can be obtained. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

The chemistry community, understandably, failed to respond at all, even though Bohmian mechanics probably holds the key to the development of a theory of chemistry, soundly based on quantum theory and relativity. The problem with molecular structure is resolved by the ontological interpretation of quantum-mechanical orbital angular momentum and the key to chemical reactivity and reaction mechanism is provided by the quantum potential function. 

Despite the interest to obtain AO integral algorithms over cartesian exponential orbitals or STO fimctions [43] in a computational universe dominated by GTO basis sets [2], this research was started as a piece of a latter project related to Quantum Molecular Similarity [44], with the concurrent aim to have the chance to study big sized molecules in a SCF framework, say, without the need to manipulate a huge number of AO functions. 

Normally the TDSE cannot be solved analytically and must be obtained numerically. In the numerical approach we need a method to render the wave function. In time-dependent quantum molecular reaction dynamics, the wave function is often represented using a discrete variable representation (DVR) [88-91] or Fourier Grid Hamiltonian (FGH) [92,93] method. A Fast Fourier Transform (FFT) can be used to evaluate the action of the kinetic energy operator on the wave function. Assuming the Hamiltonian is time independent, the solution of the TDSE may be written... 

At even lower temperatures, some unusual properties of matter are displayed. Consequently, new experimental and theoretical methods are being created to explore and describe chemistry in these regimes. In order to account for zero-point energy effects and tunneling in simulations, Voth and coworkers developed a quantum molecular dynamics method that they applied to dynamics in solid hydrogen. In liquid helium, superfluidity is displayed in He below its lambda point phase transition at 2.17 K. In the superfluid state, helium s thermal conductivity dramatically increases to 1000 times that of copper, and its bulk viscosity drops effectively to zero. Apkarian and coworkers have recently demonstrated the disappearance of viscosity in superfluid helium on a molecular scale by monitoring the damped oscillations of a 10 A bubble as a function of temperature. These unique properties make superfluid helium an interesting host for chemical dynamics. 

The quantum partition function, eq.(35), of a system can be obtained through either Monte Carlo or molecular dynamics simulations. In PIMD simulation, the corresponding Lagrangian for such a classical isomorphic system can be given as ... 

Molecular bodies of quantum mechanical electron distributions or some other molecular functions such as electrostatic potentials can be represented on various levels of approximation. These representations have two main components the physical property or model used to define a formal molecular body, and the geometrical or topological method used to describe and analyze the model. If a representation of the molecular body is selected, then the boundaries of these approximate molecular bodies can be regarded as formal molecular surfaces. Hence, the molecular shape analysis problem can be formulated as the shape analysis problem of formal molecular surfaces. 

Eq. (58) has been derived by Yomosa (1974) and can be demonstrated in different ways, see Tomasi and Persico (1994) for some derivations. This fact has been commented by several authors. The reader is referred to the clear exposition given by Tapia (1982). By comparing eq.(56) with eq.(58), one immediately recognizes that the functional J is exactly Gei, being <=1. This fact may be exploited to elaborate a direct procedure to compute Gei-If we confine ourselves to the case of an SCF description of the problem (but extension to other levels of the quantum molecular theory is immediate) the variational optimization process may be reduced to the solution of the following HF equation (which is expressed, for convenience, in a matricial form)... 

As to the content of Volume 25, the Editors thank the authors for their contributions, which give an interesting picture of part of the current state of the art of the quantum theory of matter From nonlinear-optical calculations, over a study of ion motion in molecular channels, a treatment of molecular integrals over Gaussian basis functions, and an investigation of soliton dynamics in franr-polyacetylene, to applications of quantum molecular similarity measures. 

Ross and Jungen (1999) have used differences between ab initio ion-core and Rydberg state potential curves to determine quantum defect functions. Thus the independent quantum defect functions associated with the atom-like Hel(°) and the molecular H 1) are, respectively, ai(R) and ai (R). 

Latterly, increasing use has also been made of Quantum Molecular Dynamics (QMD), based on the pioneering work of Car and Parrinello (1985) (see Chapter 8). The Car-Parrinello method makes use of Density Functional Theory to calculate explicitly the energy of a system and hence the interatomic forces, which are then used to determine the atomic trajectories and related dynamic properties, in the manner of classical MD. As an ab initio technique, QMD has the advantage over classical simulation methods that it is not reliant on interatomic potentials, and should in principle lead to far more accurate results. The disadvantage is that it demands far greater computing resources, and its application has thus far been limited to relatively simple systems. 

This problem is related to the question of appropriate electronic degeneracy factors in chemical kinetics. Whereas the general belief is that, at very low gas pressures, only the electronic ground state participates in atom recombination and that, in the liquid phase, at least most of the accessible states are coupled somewhere far out on the reaction coordinate, the transition between these two limits as a function of solvent density is by no means understood. Direct evidence for the participation of different electronic states in iodine geminate recombination in the liquid phase comes from picosecond time-resolved transient absorption experiments in solution [40, 44] that demonstrate the participation of the low-lying, weakly bound iodine A and A states, which is also taken into account in recent mixed classical-quantum molecular d5mamics simulations [42. 43]. 

W. H. Miller, J. Chem. Phys. 97 2499 (1992). (c) W. H. Miller and T. Seideman, Cumulative and state-to-state reaction probabilities via a discrete variable representation— absorbing boundary condition Green s function, Time Dependent Quantum Molecular Dynamics Experiments and Theory (J. Broeckhove, ed., NATO ARW. (d) W. H. Miller, Accts. Chem. Res. 26 174 (1993). 

An early advance in the study of the connection between electronic density and nuclear interactions was made by Parr, Gadre, and Bartolotti, followed by Parr and Berk, who pointed out the striking similarities between quantum-chemical electronic densities of molecules and another, simple molecular function that can be easily obtained from an essentially classical model based solely on the nuclear arrangement the composite nuclear potential. (In the original work of Parr and Berk the term bare nuclear potential was used.) In what follows, the basic relations between electronic density and the composite nuclear potential will be reviewed. 

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