Nuclear attraction

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

It remains to specify the elements of the one-electron core Hamiltonian, Hj y, containing the kinetic energy and nuclear attraction integrals. 

Physically, the strength of the spring representing the bond is affected by a subtle balance of nuclear repulsions, electron repulsions and electron-nuclear attractions. None of these is affected by nuclear mass and, therefore, k is not affected by isotopic substitution. 

The first term corresponds to electron-nuclear attraction, the second to electron-electron repulsion, and the third to nuclear-nuclear repulsion. 

The operator hi is a one-electron operator, representing the kinetic energy of an electron and the nuclear attraction. The operators J and K are called the Coulomb and exchange operators. They can be defined through their expectation values as follows. 

The first term on the right-hand side is a contribution from external fields, usually zero. The second term is the contribution from the kinetic energy and the nuclear attraction. The third term is the Coulomb repulsion between the electrons, and the final term is a composite exchange and correlation term. 

The P matrix involves the HF-LCAO coefficients and the hi matrix has elements that consist of the one-electron integrals (kinetic energy and nuclear attraction) over the basis functions Xi - Xn - " h matrix contains two-electron integrals and elements of the P matrix. If we differentiate with respect to parameter a which could be a nuclear coordinate or a component of an applied electric field, then we have to evaluate terms such as... 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

Calculate the ratio of the number of electrons in a neutral xenon atom to the number in a neutral neon atom. Compare this number to the ratio of the atomic volumes of these two elements. On the basis of these two ratios, discuss the effects of electron-electron repulsions and electron-nuclear attractions on atomic size. 

Here ho is the kinetic energy and nuclear attraction operator while and 1C are the coulomb and exchange operators, respectively. The coefficients X and Y are solutions of the RPA equations, which for the / singlet transition with excitation energy can be written as... 

Fh p) = Ec p) + Fxc p) + FM + FEAPhFT p) (3.15) (with subscripts C, XC, eN, Ext, and T denoting Coulomb, exchange-correlation, electron-nuclear attraction, external, and kinetic energies respectively). It is CTucial to remark that (3,15) is not the Kohn-Sham decomposition familiar in conventional presentations of DFT. There is no reference, model, nor auxiliary system involved in (3.15). From the construction presented above it is clear that in order to maintain consistency and to define functional derivatives properly all... 

Although they both have the s p valence configurations, selenium s least stable electrons are in orbitals with a larger ft value. Orbital size increases with tt. Selenium also has a greater nuclear charge than sulfur, which raises the possibility that nuclear attraction could offset increased tt. 

The most stable shape for any molecule maximizes electron-nuclear attractive interactions while minimizing nuclear-nuclear and electron-electron repulsions. The distribution of electron density in each chemical bond is the result of attractions between the electrons and the nuclei. The distribution of chemical bonds relative to one another, on the other hand, is dictated by electrical repulsion between electrons in different bonds. The spatial arrangement of bonds must minimize electron-electron repulsion. This is accomplished by keeping chemical bonds as far apart as possible. The principle of minimizing electron-electron repulsion is called valence shell electron pair repulsion, usually abbreviated VSEPR. 

Although nuclides with mass numbers around 60 are the most stable, the balance of electrical repulsion and strong nuclear attraction makes many combinations of protons and neutrons stable for indefinite times. Nevertheless, many other combinations decompose spontaneously. For example, all hydrogen nuclides with j4 > 2 are so... 

To that end, we will start with the same equation as the one used above in the case of poly electronic atoms, v/z. the eq.(lO), and we will try to use the equivalent of the compensation between the kinetic energy and the nuclear attraction (T and -Z/r) found in the atomic case. 

In fact, it turns out that the compensation between the kinetic energy and the nuclear attraction does lead to a qualitative description of the optimum orbitals in molecular systems, but only in the frame of the following restrictive conditions. 

Nuclear attraction, electron-electron repulsion, and exchange terms. 

Nuclear attraction, electron-electron repulsion and exchange terms. Using the Eqs. 6 and 7, these contributions are respectively written in terms of the quantity Wij(q) previously defined in Eq. 11 ... 

When expressing the nuclear attraction integrals in the FSGO basis, one has explicitly ... 

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... 

The 15 trivalent lanthanide, or/ -block, ions La3+, Ce3+, Pr3+, Nd3+, Pm3+, Sm3+, Eu3+, Gd3+, Tb3+, Dy3+, Ho3+, Er3+, Tm3+, Yb3+, and Lu3+, which may be collectively denoted Ln3+, represent the most extended series of chemically similar metal ions. The progressive filling of the 4/orbitals from La3 + to Lu3 + is accompanied by a smooth decrease in rM with increase in atomic number as a consequence of the increasingly strong nuclear attraction for the electrons in the diffuse / orbitals (the lanthanide contraction). Thus, the nine-coordinate rM decrease from 121.6 to 103.2 pm from La3+ to Lu3+, and the eight-coordinate ionic radii decrease from 116.0 to 97.7 pm from La3+ to Lu3+ (2). Ligand field effects are small by comparison with those observed for the first-... 

In the latter expression, V(r,) is the electron-nuclear attraction operator describing the interaction of the z th electron of the molecule with the set of nuclei. 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

As for atoms, it is assumed that if the wave function for a molecule is a single product of orbitals, then the energy is the sum of the one-electron energies (kinetic energy and electron-nuclear attractions) and Coulomb interactions... 

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