Molecular wave equation

There is no mystery here, only misleading terminology. The so-called "molecular" wave equation in fact describes all possible distributions of the a + i particles and a specific solution can only be the result of an equally specific set of boundary conditions, which depend not only on the physical environment of the system, but also on the history of the particles as a set. It is well known that, without exception, the constituent fragments of a molecule may exist, without change, under exactly the same conditions as... 

An alternative strategy is to synthesize a molecular wave function, on chemical intuition, and progressively modify this function until it solves the molecular wave equation. However, chemical intuition fails to generate molecular wave functions of the required spherical symmetry, as molecules are assumed to have non-spherical three-dimensional structures. The impasse is broken by invoking the Born-Oppenheimer assumption that separates the motion of electrons and nuclei. At this point the strategy ceases to be ab initio and reduces to semi-empirical quantum-mechanical simulation. The assumed three-dimensional nuclear framework is no longer quantum-mechanically defined. The advantage of this model over molecular mechanics is that the electron distribution is defined quantum-mechanically. It has been used to simulate the H2 molecule. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

Treatment of heteronuclear biatomic molecules by LCAO-MO theory is similar to that of the homonuclear species. The values of mixture coefficients Ci and C2 in the molecular wave equation... 

The Bom-Oppenheimer approximation is used to separate out the electronic Schrodinger equation from the complete molecular wave equation. Throughout this discussion, tl> will represent the electronic component of the total wavefunction, which is represented as... 

The function III. 120 with more general forms of the functions u and g has also been studied in greater detail by Baber and Hasse (1937) and by Pluvinage (1950). The latter expanded g(rl2) in a power series in r12 and, by studying the formal properties of the wave equation itself, Pluvinage could derive certain general relations for the coefficients. At the Paris molecular symposium in 1957, Roothaan reported that, by expressing u and g in the form... 

After the discovery of quantum mechanics in 1925 it became evident that the quantum mechanical equations constitute a reliable basis for the theory of molecular structure. It also soon became evident that these equations, such as the Schrodinger wave equation, cannot be solved rigorously for any but the simplest molecules. The development of the theory of molecular structure and the nature of the chemical bond during the past twenty-five years has been in considerable part empirical — based upon the facts of chemistry — but with the interpretation of these facts greatly influenced by quantum mechanical principles and concepts. 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

Molecular mechanics (also known diS force-field calculations) is a method for the calculation of conformational geometries. It is used to calculate bond angles and distances, as well as total potential energies, for each conformation of a molecule. Steric enthalpy can be calculated as well. Molecular orbital calculations (p. 34) can also give such information, but molecular mechanics is generally easier, cheaper (requires less computer time), and/or more accurate. In MO calculations, positions of the nuclei of the atoms are assumed, and the wave equations take account only of... 

Equation (2) was also used to calculate quantum chemical approach. On the basis of previous results [19], calculated electrostatic potentials were computed from ab initio wave functions obtained in the framework of the HF/SCF method using a split-valence basis set (3-21G) and a split-valence basis set plus polarisation functions on atoms other than hydrogen (6-31G ). The GAUSSIAN 90 software package [20] was used. Since ab initio calculations of the molecular wave function for the whole... 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

In this equation, f is the molecular wave function, is an atomic wave function, and a is a weighting coefficient that gives the relative weight in the "mix" of the atomic wave functions. The summation is... 

Having shown that the weighting coefficient (A) of the term giving the contribution of an ionic structure to the molecular wave function is related to the dipole moment of the molecule, it is logical to expect that equations could be developed that relate the ionic character of a bond to the electronegativities of the atoms. Two such equations that give the percent ionic character of the bond in terms of the electronegativities of the atoms are... 

Although the equations look very different, the calculated values for the percent ionic character are approximately equal for many types of bonds. If the difference in electronegativity is 1.0, Eq. (3.70) predicts 19.5% ionic character while Eq. (3.71) gives a value of 18%. This difference is insignificant for most purposes. After one of these equations is used to estimate the percent ionic character, Eq. (3.61) can be used to determine the coefficient A in the molecular wave function. Figure 3.10 shows how percent ionic character varies with the difference in electronegativity. 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

The molecular wave function can then be written as a product of the four wave functions of Equations 2.8 through 2.11 while the total energy Emoi can be expressed as a sum of four energy terms. 

We can compute from first principles all possible vibrational modes for 3iA oscillators in the cell unit, solving the Schrodinger equation with appropriate atomic (and/or molecular) wave functions. 

A procedure for obtaining an approximate wave function of a molecular orbital by treating the molecule as different canonical forms. The overall wave expression is thus the weighted sum of the wave equation for each of these canonical forms. 

The molecular wave function is a linear combination ip = api +bp2, with coefficients a and b to be determined. Because the calculations are independent of the spin coordinates, we omit them. To set up the secular equation we must evaluate the matrix elements Hij and Sij. Because pi and p2 are normalized, = 1 = 822- Because the basis functions are really two copies of the same function - only centered at different positions in space - some of the matrix elements are equal Hu = H22, H12 = H21, = S22 = 1), and... 

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