Spectroscopic accuracy

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

If affordable, there is a range of very accurate coupled-cluster and symmetry-adapted perturbation theories available which can approach spectroscopic accuracy [57, 200, 201]. However, these are only applicable to the smallest alcohol cluster systems using currently available computational resources. Near-linear scaling algorithms [192] and explicit correlation methods [57] promise to extend the applicability range considerably. Furthermore, benchmark results for small systems can guide both experimentalists and theoreticians in the characterization of larger molecular assemblies. 

In order to obtain spectroscopic accuracy it was found necessary, as discussed in Chapter 4, to introduce higher-order terms, for example, products and powers of Casimir operators. These higher-order terms can be dealt with easily within the approach discussed here. For example, a Hamiltonian of the type... 

The accuracy/cost ratio of conventional multideterminantal methods depends on two factors. The first, the correlation or n-particle factor (discussed earlier), can be successfully reduced via the rapidly converging coupled-duster ansatz [see Eq. (13.4)]. The second, file one-partide factor, is the primary reason why predictions with spectroscopic accuracy using conventional expansion methods are hardly feasible even for the helium... 

Truly spectroscopic accuracy for total energies is still difficult to obtain from either explicitly correlated methods or extrapolation approaches for all but the simplest... 

Another motivation for considering molecular systems without assuming the BO approximation stems from the realization that in order to reach spectroscopic accuracy in quantum-mechanical calculations (i.e., error less than 1 microhartree), one needs to account for the coupling between motions of electrons and nuclei and, in some cases, also for the relativistic effects. Modern experimental techniques, such as gas-phase ion-beam spectroscopy, reach... 

The vast majority of quantum chemical studies focus on equilibrium properties. However, a detailed understanding of chemical reactions requires a description of their chemical dynamics, which in turn requires information about the change in potential energy as bonds are broken or formed. Even though modem electronic structure theory can provide near-spectroscopic accuracy for small molecular systems near their equilibrium geometries, the general description of potential energy surfaces away from equilibrium remains very much a frontier area of research. 

The potential curves derived from such calculations can often be empirically improved by comparison with so-called experimental curves derived from observed spectroscopic data, using Rydberg-Klein-Rees (RKR) or other inversion procedures. It is often found, particularly for the atmospheric systems, that the remaining correlation errors in a configuration interaction (Cl) calculation are similar for many excited electronic states of the same symmetry or principal molecular-orbital description. Thus it is often possible to calibrate an entire family of calculated excited-state potential curves to near-spectroscopic accuracy. Such a procedure has been applied to the systems described here. 

Application of HY-CI and other methods to simple atomic and molecular systems is also discussed in dementi, et a/.[68]. In particular, these authors give examples of computational times on a mainframe computer. They estimate that carrying out calculations on H3 with a modest HY-CI basis would require from hundreds to thousands of hours to evaluate the integrals alone, while a larger calculation attempting to match spectroscopic accuracy would approach times more often associated with geological processes than with quantum chemistry. 

Rather few papers have dealt with the computation of thermodynamic functions from the results of ab initio calculations, but for H2, where the latter are of spectroscopic accuracy, Kosloff, Levine, and Bernstein have computed the thermodynamic properties of Ha, Da, and HD, using the best theoretical results.75 This work represents the first example of an accurate determination of a bulk, macroscopic property from first principles. 

The methods described in Sect. 3 for the calculation of accurate nonrelativistic wave functions and energies can in principle be applied to more complex atoms and molecules. The principal difficulties are that the number of terms required in the basis set to reach a given level of accuracy grows extremely rapidly with the number of particles, and the correlated integrals become much more difficult to evaluate. Only in the case of lithium (and Li-like ions) have results of spectroscopic accuracy been obtained (see Ref. [69] for a review). However, the demand on computer resources increases by about a factor of 6000 to reach the same level of accuracy. 

Groenenboom G, Van der Avoird A, Wormer PES, Mas EM, Bukowski R, Szalewicz K (2000) Water pair potential of near spectroscopic accuracy. II. Vibration-rotation-tunneling levels of the water dimer. J Chem Phys 113 6702—6715... 

It should be noted that the goal of true spectroscopic accuracy may be unattainable because of the implicit errors associated with the use of a Born-Oppenheimer, nonrelativistic Hamiltonian to describe molecular systems (Ref. 225). 

In the first part of this article we have briefly summarized the study of resonant dipole-dipole energy transfer collisions between Rydberg atoms. Due to the long range of the dipole-dipole interaction the collision process can be understood with nearly spectroscopic accuracy, and this understanding forms the basis for understanding dipole-dipole interactions in a frozen Rydberg gas. 

Density functional theory (DFT) [62] incorporates electron correlation at a very small computational cost, but its suitability for weak interaction still seems a somewhat open question. There have been comparisons of DFT and conventional methods [63 68]. Mostly, an improvement over SCF level treatment seems possible, but there is a clear dependence on the choice of functional and on basis-set size. Also, DFT may be more sensitive to BSSE in smaller basis sets than conventional treatments [64]. A single functional choice for spectroscopic accuracy in treating weakly bound clusters does not yet seem at hand, but that alone does not preclude application of DFT for lower levels of accuracy. With the computational cost advantage of DFT, the capability exists for treating large, extended clusters. 

Because of the use of N distinct sets, we expect the spin-coupled VB expansion to converge much faster, and indeed the results so far show that this is the case. As we discuss in more detail below, the spin-coupled function (18) by itself possesses all the correct qualitative characteristics of the ground state of a molecular system, and 200-700 terms of expansion (17) are sufficient to attain chemical accuracy ( 0.1 eV) for the first 10-15 eigenstates of a given symmetry, and spectroscopic accuracy ( 100 cm ) for the ground state. ° ... 

Strictly speaking, a potential function for a molecule is purely a theoretical intermediate in the approximate calculation of the energy levels and not experimentally observable. There would seem to be little reason, then, to speak of an experimental potential function. On the other hand, it is impractical, if not impossible, to compute with spectroscopic accuracy the theoretical potential functions for all molecules of physical interest. Thus it is not surprising that numerous methods have been developed for estimating the potential function from experimental data. [Davidson, 1962]... 

The Coupled-clusters (CC) method[7] based on the cluster expansion of the wavefunction has been established as a highly reliable method for calculations of ground state properties of small molecules with the spectroscopic accuracy. When this method is used together with a flexible basis set it recovers the dominant part of the electron correlation. Typically, CC variant explicitly considering single and double excitations (CCSD) is used. In order to save computer time the contributions from triple excitations are often calculated at the perturbation theory level (notation CCSD(T) is used in this case). CCSD(T) method can be routinely used only for systems with about 10 atoms at present. Therefore, it cannot be used directly in zeolite modeling, however, results obtained at CCSD(T) level for small model systems can serve as an important benchmark when discussing the reliability of more approximate methods. 

Our goal is to obtain a general method for computing bound state as well as scattering properties of general few-body systems. The need for experimental ( sometimes spectroscopic ) accuracy requires us to be able to describe fundamental few-body systems in detail without making formal as well as uncontrollable numerical approximations. 

CC-VSCF was also used for computing anharmonic vibrational spectra from DFT potentials [103,104], The most accurate spectroscopic results are obtained from the hybrid functionals B97 [105] and B3LYP [106], of the non-hybrid DFT functionals, HCTH [107] seems superior to BLYP [108], The spectroscopic accuracy of B97 and B3LYP was about equal to that of MP2. 

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