Orbital hydrogen-like

Millikan has shown that the overlap integral for hydrogen-like p orbitals in linear hydrocarbons is about 0.27 (Millikan, 1949). 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Orbital Surfaces. Molecular orbitals provide important clues about chemical reactivity, but before we can use this information we first need to understand what molecular orbitals look like. The following figure shows two representations, a drawing and a computer-generated picture, of a relatively high-energy, unoccupied molecular orbital of hydrogen molecule, H2. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

We know from Section 1.5 that cr bonds are cylindrically symmetrical. In other words, the intersection of a plane cutting through a carbon-carbon singlebond orbital looks like a circle. Because of this cjdindrical symmetry rotation is possible around carbon-carbon bonds in open-chain molecules. In ethane, for instance, rotation around the C-C bond occurs freely, constantly changing the spatial relationships between the hydrogens on one carbon and those on the other (Figure 3.5),... 

Different Types of Complete Sets. Importance of the Continuum in Using Hydrogen-like Orbitals... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

The discrete hydrogen-like orbitals (nlm) are given by the formula ... 

The first excited orbitals are characterized by being localized mainly within the same region of space as Xv The function xni has (n—l 1) nodes but is otherwise not particularly hydrogen-like it may be expanded in the standard hydrogen-like functions only if a considerable contribution from the continuum is included. 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as... 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

The radial wave functions used are thus the hydrogen-like 2p and 3d functions, J ai(r) and J 32-(r), for all orbitals of the L and M shells, respectively the symbols pts, and i 3j, piP, p3d represent these multiplied by the angular parts 1 (for s), /3 cos 8 (for p), and /5/4 (3 cos2 0-1) (for d), rather than the usual hydrogen-like orbitals. The 2-axis for each atom points along the internuclear axis toward the other atom. 

It might be thought that these values of y are not correct because of the fact that the electron shells actually do not consist of hydrogen-like electrons, but rather themselves of penetrating electrons. However, as Z increases the penetrating orbits become more and more hydrogen-like and these... 

To evaluate the segmentary quantum numbers we observe from a comparison of equations (9) and (10) with the corresponding ones for a hydrogen-like orbit... 

Now many physical properties depend mainly on the behaviour of the electron in the outer part of its orbit. As an example we may mention the mole refraction or polarizability of an atom, which arises from deformation of the orbit in an external field. This deformation is greatest where the ratio of external field strength to atomic field strength is greatest that is, in the outer part of the orbit. Let us consider such a property which for hydrogen-like atoms is found to vary with nrZ t. Then a screening constant for this property would be such that... 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

The main aspect of the eq.(17) is that the orbital energy e occurs only in the coefficients Uk with k > 2. Therefore we obtain here the same results as the one obtained in the case of hydrogen-like atoms ( 1.1 and 1.2) ... 

In the asymptotic region, an electron approximately experiences a Z /f potential, where Z is the charge of the molecule-minus-one-electron ( Z = 1 in the case of a neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the ip orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrodinger equation for the hydrogen-like atom with atomic number Z. ... 

We present in the Table 2 the ratio of the irregular solution of the hydrogen-like system with the s wave of the optimised orbital, and with the s wave of the unoptimised orbital. It is seen that the irregular numerical solution is actually much closer to be proportional to the s wave of the optimised orbital than to that of the unoptimised orbital. 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

Consider a crude approximation to the ground state of the lithium atom in which the electron-electron repulsions are neglected. Construct the ground-state wave function in terms of the hydrogen-like atomic orbitals. 

The ground-state wave function for the unperturbed two-electron system is the product of two Is hydrogen-like atomic orbitals... 

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