Wavefunctions molecules

Table 2 Electric field vector E at the nuclei in ethene and ethyne,11 calculated using a large gaussian basis SCF-MO wavefunction (molecules in xy-plane, C—C axis is y-axis). The atomic unit of electric field is el4ne0a02x 5.142 x 10u V m-1...

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

Calculating the Energy from the Wavefunction the Hydrogen Molecule... 

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

Ale now substitute the hydrogen molecule wavefunction into Equation (2.73) to provide the ollowing ... 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

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 ... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

For homonuclear molecules (e.g., O2, N2, etc.) the inversion operator i (where inversion of all electrons now takes place through the center of mass of the nuclei rather than through an individual nucleus as in the atomic case) is also a valid symmetry, so wavefunctions F may also be labeled as even or odd. The former functions are referred to as gerade (g) and the latter as ungerade (u) (derived from the German words for even and odd). The g or u character of a term symbol is straightforward to determine. Again one... 

Finally, for linear molecules in Z states, the wavefunctions can be labeled by one additional quantum number that relates to their symmetry under reflection of all electrons through a ay plane passing through the molecule s Coo axis. If F is even, a + sign is appended as a superscript to the term symbol if F is odd, a - sign is added. 

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