Core Potentials

There are several variations of this method. The PRDDO/M method is parameterized to reproduce electrostatic potentials. The PRDDO/M/FCP method uses frozen core potentials. PRDDO/M/NQ uses an approximation called not quite orthogonal orbitals in order to give efficient calculations on very large molecules. The results of these methods are fairly good overall, although bond lengths involving alkali metals tend to be somewhat in error. 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

There are several issues to consider when using ECP basis sets. The core potential may represent all but the outermost electrons. In other ECP sets, the outermost electrons and the last filled shell will be in the valence orbital space. Having more electrons in the core will speed the calculation, but results are more accurate if the —1 shell is outside of the core potential. Some ECP sets are designated as shape-consistent sets, which means that the shape of the atomic orbitals in the valence region matches that for all electron basis sets. ECP sets are usually named with an acronym that stands for the authors names or the location where it was developed. Some common core potential basis sets are listed below. The number of primitives given are those describing the valence region. 

SBKJC VDZ Available for Li(4.v4/>) through Hg(7.v7/ 5d), this is a relativistic basis set created by Stevens and coworkers to replace all but the outermost electrons. The double-zeta valence contraction is designed to have an accuracy comparable to that of the 3—21G all-electron basis set. Hay-Wadt MB Available for K(5.v5/>) through Au(5.v6/ 5r/), this basis set contains the valence region with the outermost electrons and the previous shell of electrons. Elements beyond Kr are relativistic core potentials. This basis set uses a minimal valence contraction scheme. These sets are also given names starting with LA for Los Alamos, where they were developed. 

Some of the basis sets discussed here are used more often than others. The STO—3G set is the most widely used minimal basis set. The Pople sets, particularly, 3—21G, 6—31G, and 6—311G, with the extra functions described previously are widely used for quantitative results, particularly for organic molecules. The correlation consistent sets have been most widely used in recent years for high-accuracy calculations. The CBS and G2 methods are becoming popular for very-high-accuracy results. The Wachters and Hay sets are popular for transition metals. The core potential sets, particularly Hay-Wadt, LANL2DZ, Dolg, and SBKJC, are used for heavy elements, Rb and heavier. 

The methods listed thus far can be used for the reliable prediction of NMR chemical shifts for small organic compounds in the gas phase, which are often reasonably close to the liquid-phase results. Heavy elements, such as transition metals and lanthanides, present a much more dilficult problem. Mass defect and spin-coupling terms have been found to be significant for the description of the NMR shielding tensors for these elements. Since NMR is a nuclear effect, core potentials should not be used. 

This chapter provides only a brief discussion of relativistic calculations. Currently, there is a small body of references on these calculations in the computational chemistry literature, with relativistic core potentials comprising the largest percentage of that work. However, the topic is important both because it is essential for very heavy elements and such calculations can be expected to become more prevalent if the trend of increasing accuracy continues. 

The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

The use of RECP s is often the method of choice for computations on heavy atoms. There are several reasons for this The core potential replaces a large number of electrons, thus making the calculation run faster. It is the least computation-intensive way to include relativistic effects in ah initio calculations. Furthermore, there are few semiempirical or molecular mechanics methods that are reliable for heavy atoms. Core potentials were discussed further in Chapter 10. 

Many researchers have performed calculations that include the two large-magnitude components of the spinnors. This provides a balance between high accuracy and making the calculation tractable. Such calculations are often done on atoms in order to obtain the wave function description used to create relativistic core potentials. 

Core potentials are seldom used for organic molecules because there are so few electrons in the core. Relativistic effects are seldom included since they have very little effect on the result. Ah initio methods are discussed further in Chapter 3. 

The heavier elements are affected by relativistic effects. This is most often accounted for by using relativistic core potentials. Relativistic effects are discussed in more detail in Chapters 10 and 33. 

The alkali metals tend to ionize thus, their modeling is dominated by electrostatic interactions. They can be described well by ah initio calculations, provided that diffuse, polarized basis sets are used. This allows the calculation to describe the very polarizable electron density distribution. Core potentials are used for ah initio calculations on the heavier elements. 

Ah initio methods pose problems due a whole list of technical difficulties. Most of these stem from the large number of electrons and low-energy excited state. Core potentials are often used for heavier elements to ease the computational requirements and account for relativistic elfects. 

Relativistic effects are significant for the heavier metals. The method of choice is nearly always relativistically derived effective core potentials. Explicit spin-orbit terms can be included in ah initio calculations, but are seldom used because of the amount of computational effort necessary. Relativistic calculations are discussed in greater detail in Chapter 33. 

Relativistic effects should always be included in these calculations. Particularly common is the use of core potentials. If core potentials are not included, then another form of relativistic calculation must be used. Relativistic effects are discussed in more detail in Chapter 33. 

Ah initio calculations with core potentials are usually the method of choice. The researcher must make a difficult choice between minimizing the CPU time requirements and obtaining more accurate results when deciding which core potential to use. Correlation is particularly difficult to include because of the large number of electrons even in just the valence region of these elements. 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

Basis sets for atoms beyond the third row of the periodic table are handled somewhat differently. For these very large nuclei, electrons near the nucleus are treated in an approximate way, via effective core potentials (ECPs). This treatment includes some relativistic effects, which are important in these atoms. The LANL2DZ basis set is the best known of these. 

Optimize these three molecules at the Hartree-Fock level, using the LANL2DZ basis set, LANL2DZ is a double-zeta basis set containing effective core potential (ECP) representations of electrons near the nuclei for post-third row atoms. Compare the Cr(CO)5 results with those we obtained in Chapter 3. Then compare the structures of the three systems to one another, and characterize the effect of changing the central atom on the overall molecular structure. 

Predict the structure and frequencies for this compound using two or more different DFT functionals and the LANL2DZ basis set augmented by diffuse functions (this basis set also includes effective core potentials used to include some relativistic effects for K and Cs). How well does each functional reproduce the observed spectral data ... 

Edit Output File icon xlix effective core potentials 101 electron affinity 142 electron correlation 6, 114,118 electron density 165 electron spin 259 electronic structure theory 3 electrostatic potential-derived charges CHelpG 196... 

Effective core potentials (ECP) replace the atomic core electrons in valence-only ab initio calculations, and they are often used when dealing with compounds containing elements from the second row of the periodic table and above. 

After some experience with MNDO, it became clear that there were certain systematic en ors. For example the repulsion between two atoms which are 2-3 A apart is too high. This has as a consequence that activation energies in general are too large. The source was traced to too repulsive an interaction in the core-core potential. To remedy this, the core-core function was modified by adding Gaussian functions, and the whole model was reparameterized. The result was called Austin Model 1 (AMl) in honour of Dewar s move to the University of Austin. The core-core repulsion of AMI has the form ... 

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