Physical Chemistry

Physical Chemistry Chemistry Department of Fudan University

Hydrogen-like Atom: the model consists of a proton fixed at the origin and an electron that interacts with the proton through a Coulombic potential. spherical coordinate Chemistry Department of Fudan University

Note that the angular and radial terms can be separated, we suggest that we can write the wavefunction as a product of radial and angular parts. : Then the angular part is separated into two parts: and three parts are substituted into Schrodinger equation, we have Chemistry Department of Fudan University

Now the Schrodinger equation can be written as three separate equations. radial equation colatitude equation azimuthal equation Chemistry Department of Fudan University

We have seen that the azimuthal wave functions are This solution imposes the constraint the m be a quantum number and have values m = 0, ±1, ±2, ±3, … When this equation is solved it is found that k must equal l(l+1) with l = 0, 1, 2, 3… and as above m = 0, ±1, ±2, ±3, … Chemistry Department of Fudan University

Chemistry Department of Fudan University

The coefficient of each power of r must be zero, so we can derive the recursion relation for the constants bk The power series must be terminated for some value of Chemistry Department of Fudan University

l = 0, 1, 2, 3… The coefficients before the terms are zero. l, l+1,….n-1 This is a power series of with terms Chemistry Department of Fudan University

Physical Significance of the Solution 1.atomic orbital There are three quantum numbers for each eigenfunction of a hydrogenlike atom. The orbitals with different quantum numbers are orthogonal. Chemistry Department of Fudan University

The wavefunctions are difficult to represent because they are complex. This problem can be solved by using linear combinations ,which are not complex. Chemistry Department of Fudan University

3. Principal quantum number n This quantum number can have any integer value from 1 up to infinity. is the energy required to take the electron from the ground state to . All the orbitals of a given value n are said to form a single shell of the atom. In hydrogen atom, all orbitals of a given n, have the same energy. Chemistry Department of Fudan University

4. Orbital quantum number l For a given value of n, this quantum number can have any integer value from 0 up to n – 1. The orbitals with the same value n but different values of l are said to form a subshell. For historical reasons, we associate letter symbols with the value of . l=0(s), 1(p), 2(d), 3(f), …... Chemistry Department of Fudan University

Magnetic quantum number ml For a given value of l, this quantum number can be any integer value starting at –l and going up to +l. Chemistry Department of Fudan University

Problem: Use hydrogenic orbitals to calculate the mean radius of a 1s orbital. A Hydrogen atom is in its 4d state. The atom decays to a lower state by emitting a photon. Find the possible photon energies that may be observed. Give your answers in eV Chemistry Department of Fudan University

Summary of solution The Application of the number of nodes Chemistry Department of Fudan University

Radial function Radial distribution Radial properties Radial probability density The radial wavefunction of some states of hydrogen atom. Chemistry Department of Fudan University

Hydrogen 2s Radial Probability Chemistry Department of Fudan University

Hydrogen 2p Radial Probability Chemistry Department of Fudan University

Hydrogen 3s Radial Probability Chemistry Department of Fudan University

Hydrogen 3p Radial Probability Chemistry Department of Fudan University

Hydrogen 3d Radial Probability Chemistry Department of Fudan University

Radial probability density r2R2 for a hydrogen atom Chemistry Department of Fudan University

Angular wavefunction & Angular probability density Boundary surfaces of the s-orbital and three (real) hydrogen p-orbitals Chemistry Department of Fudan University

Boundary surfaces of the five (real) hydrogen d-orbitals Chemistry Department of Fudan University

Boundary surfaces of the seven (real) hydrogen f-orbitals Chemistry Department of Fudan University

Contour map of the 2s atomic orbital and its charge density distributions for the H atom. The zero contours shown in the maps for the orbitals define the positions of the nodes. Negative values for the contours of the orbital is indicated by dashed lines, positive values by solid lines Chemistry Department of Fudan University

Contour map of the 2p atomic orbital and its charge density distributions for the H atom. The zero contours shown in the maps for the orbitals define the positions of the nodes. Negative values for the contours of the orbital is indicated by dashed lines, positive values by solid lines Chemistry Department of Fudan University

Contour map of the 3d atomic orbital and its charge density distributions for the H atom. The zero contours shown in the maps for the orbitals define the positions of the nodes. Negative values for the contours of the orbital is indicated by dashed lines, positive values by solid lines Chemistry Department of Fudan University

Contour map of the 4f atomic orbital and its charge density distributions for the H atom. The zero contours shown in the maps for the orbitals define the positions of the nodes. Negative values for the contours of the orbital is indicated by dashed lines, positive values by solid lines. Chemistry Department of Fudan University

Multi-electron Atoms General form of Hamiltonian for an k-electron atom Atomic Unit Chemistry Department of Fudan University

Atomic Units Clearly, if we work with energy in SI units in atomic and molecular calculation, we are in danger of underflows or overflows. Therefore, we need an additional modification of the units. • Bohr radius: atomic unit of length a0 = 0.5291772083x10-10m • Hartree: the Coulomb repulsion between two electrons separated by one bohr Eh = 4.35974381x10-18J • atomic unit of mass: mu = 1.66053873x10-27kg • atomic unit of charge: e = 1.602176462x10-19Catomic unit of force : Eh/a0 = 8.23872181x10-8Natomic unit of time: /Eh = 2.418884326500x10-17s • atomic unit of momentum: /a0 = 1.99285151x10-24kgms-1 Chemistry Department of Fudan University

Hartree The One-Electron Approximation: assume that the Hamiltonian is a sum of one-electron functions, with an approximate potential energy that takes the average interaction of the electrons into account Chemistry Department of Fudan University

Not only the one-electron approximation permits us to separate the many-electron schrodinger equation, but it also makes the solution of the resulting equation trivial. Chemistry Department of Fudan University

Central-field Approximation Slater Chemistry Department of Fudan University

effective nuclear charge Since the electron on the average experiences a reduced nuclear charge (i.e., the effective nuclear charge) because of the screening effect of the second electron, the size of the orbital should be determined by an effective nuclear charge, rather than by the actual nuclear charge. Chemistry Department of Fudan University

Shielding and Penetration An electron in a many-electron atom experiences a repulsion that can be represented by a point negative charge located at the nucleus and equal in magnitude to the total charge of the electrons within a sphere of radius. The effect of this point negative charge, is to reduce the full charge of the nucleus from Ze to Zeffe. The difference between Z and Zeff is called the shielding constant. The screening constant for a given subshell is the sum of contributions from other electrons in the atom. Chemistry Department of Fudan University

Semiempirical Methods Self-consistent Field Method Some approximation forms of wavefunction were guessed as the first order trial function and were substituted into Schrodinger equation. Then the equation can be solved numerically. The calculation gives the form of wavefunctions. In general they will differ from the set used initially. These improved orbitals are used in another cycle of calculation until the orbitals and energies obtained are insignificantly different from those used at the start of the current cycle. Chemistry Department of Fudan University

Electron Spin Experimental Results D-line of sodium consists of two lines with 589.6nm and 589.0nm. Chemistry Department of Fudan University

Hydrogen atoms was directed through a strong inhomogeneous magnet. The atomic beam was split into two components Chemistry Department of Fudan University

Since the spin angular momentum of an electron has no analog in classical mechanics, we cannot construct spin angular momentum operators by first writing the classical Hamiltonian. A complete wavefunction for an atom must indicate the spin state of the electron. Chemistry Department of Fudan University

The treatment of spin angular momentum is closely analogous to the treatment of orbital angular momentum. Some properties of electron spin Chemistry Department of Fudan University

The spin eigenfunctions are orthonormal Thus, a complete state specification for an atom requires four quantum numbers Chemistry Department of Fudan University

Problem: The spin functions and can be expressed as and The spin operator can be represented by Show that and . Chemistry Department of Fudan University

2 2 a 1 1 b Antisymmetric function Symmetric Function Chemistry Department of Fudan University

symmetric nonsymmetric symmetric antisymmetric Chemistry Department of Fudan University

The wavefunction for any system of electrons must be antisymmetric with respect to the interchange of any two electrons Slater determinant No two electrons in any atom have the same four quantum numbers. Chemistry Department of Fudan University

Problem: Show that the Slater determinants for Helium atom and Lithium atom. Chemistry Department of Fudan University

Physical Chemistry