Chemistry Department of Fudan University Physical Chemistry 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Now the Schrodinger equation can be written as three separate equations. radial equation colatitude equation azimuthal equation 2019/2/22 Chemistry Department of Fudan University

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, … 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 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 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

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), …... 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Summary of solution The Application of the number of nodes 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Radial properties Radial function Radial distribution Radial probability density The radial wavefunction of some states of hydrogen atom. 2019/2/22 Chemistry Department of Fudan University

Hydrogen 2s Radial Probability 2019/2/22 Chemistry Department of Fudan University

Hydrogen 2p Radial Probability 2019/2/22 Chemistry Department of Fudan University

Hydrogen 3s Radial Probability 2019/2/22 Chemistry Department of Fudan University

Hydrogen 3p Radial Probability 2019/2/22 Chemistry Department of Fudan University

Hydrogen 3d Radial Probability 2019/2/22 Chemistry Department of Fudan University

Radial probability density r2R2 for a hydrogen atom 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Angular wavefunction & Angular probability density Boundary surfaces of the s-orbital and three (real) hydrogen p-orbitals 2019/2/22 Chemistry Department of Fudan University

Boundary surfaces of the five (real) hydrogen d-orbitals 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Boundary surfaces of the seven (real) hydrogen f-orbitals 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Multi-electron Atoms General form of Hamiltonian for an k-electron atom Atomic Unit 2019/2/22 Chemistry Department of Fudan University

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 = 1.602176462x10-19C atomic unit of force : Eh/a0 = 8.23872181x10-8N atomic unit of time: /Eh = 2.418884326500x10-17s atomic unit of momentum: /a0 = 1.99285151x10-24kgms-1 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

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 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Central-field Approximation Slater 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 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. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Electron Spin Experimental Results D-line of sodium consists of two lines with 589.6nm and 589.0nm. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Hydrogen atoms was directed through a strong inhomogeneous magnet. The atomic beam was split into two components 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

Some properties of electron spin The treatment of spin angular momentum is closely analogous to the treatment of orbital angular momentum. Some properties of electron spin 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University The spin eigenfunctions are orthonormal Thus, a complete state specification for an atom requires four quantum numbers 2019/2/22 Chemistry Department of Fudan University

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 . 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 1 2 a b 1 2 Symmetric Function Antisymmetric function 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University symmetric nonsymmetric antisymmetric symmetric 2019/2/22 Chemistry Department of Fudan University

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. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Problem: Show that the Slater determinants for Helium atom and Lithium atom. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Building Principle The order of occupation is: Pauli Exclusion Principle: in an atom no two electrons can have all four quantum numbers the same. Aufbau ( Buliding Up) Principle: At the ground state of the atom, electrons will occupy the lowest energy orbitals first, and only fill the higher energy orbitals when no lower energy orbitals are left. Hund principle: Electrons occupy different orbitals of a given subshell before doubly occupying any one of them 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Periodic Table The electron configurations (or atomic mass ?) of the elements account for the periodicity of physical and chemical properties of the elements. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Ionization energy: the energy required to remove an electron completely from a gaseous atom, molecule, or ion. Eigen-energy of orbital: the energy obtained from the single electron schrodinger equation. Electron affinity: the energy released in the process of adding an electron to an atom. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Problem: 1.Estimate the effective nuclear charge for a 1s electron in He, if the first ionization energy of helium is 24.6eV. 2. Estimate the effective nuclear charge felt by the 2s electron in the lithium atom, if the ionization energy is 5.83eV. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Correlation Effect of Spin What is the difference in energy after the electronic spin is considered? 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University electronic spin is considered Electronic spin is neglected Hund rule 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Atomic Term Symbols The state of the multi-electron atom The arrangement of electrons in an atom is known as its electronic configuration. What is the coupling of the angular Momentum 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University - coupling: the orbital momenta for all electrons is coupled to obtain a total orbital angular momentum L and then the individual spin angular momenta is coupled to obtain the total spin angular momentum S. The total angular momentum is then obtained by vector addition of L and S. - coupling: the orbital angular momentum l and spin angular momentum s of each electron are coupled to obtain a total angular momentum j. The all the individual j are coupled to obtain the total angular momentum J of an atom. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Quantum Numbers of Atom 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Quantum Numbers of Atom 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Spin multiplicity Atomic Term Symbol Atomic Levels If the Pauli principle does not have to be considered, we can write all the term states directly. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University L=0, S=0 1S singlet L=2, S=1 3D triplet Number of possible configurations f7 max Count the total ms S max L Count the total ml 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Atomic terms symbols for two non-equivalent electrons pp 3D 1D 3P 1P 3S 1S Atomic terms symbols for two equivalent electrons p2 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 1. The term with maximum multiplicity lies lowest in energy. 2. For a given multiplicity, the term with the largest value of L lies lowest in in energy. Why C 3P0 3. For atoms with less than half-filled shells, the level with the lowest value of J lies lowest in energy. If the subshell is more than half filled, the state with maximum J is lowest in energy. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Problem: Show the atomic term symbols for Helium and Nitrogen in their ground states. What is the spectroscopic term of the ground state of the Li atom? If the 2s electron is excited to the 2p state, what terms are then possible? 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Addition of Angular momentum: e.g 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University ms=1 S=1 ms=-1 triplet + S=1 ms=0 S=0 ms=0 singlet 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Problem: Show the atomic term symbols for Helium and Nitrogen in their ground states. What is the spectroscopic term of the ground state of the Li atom? If the 2s electron is excited to the 2p state, what terms are then possible? 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University One electron atom Summary L-S coupling Magnetic field 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Perturbation Theory The Hamiltonian is partitioned into two parts: one is the zeroth-order Hamiltonian, and the other is the perturbation. The zeroth-order Schrodinger equation can be solved exactly. It is then assumed that the perturbation is sufficiently small that its presence does not appreciably alter the eigen-functions for the system. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University In general, perturbed Energy and WF can be obtained by using Taylor expansion to treat each one-electron Energy and WF Typically, a second order of approximation will be accurate enough, but the calculation is already very difficult 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Example: Using the first-order perturbation theory to compute the ground-state energy of a helium atom The zeroth-order Hamiltonian 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University The measured energy of the helium atom is -79.01eV. The first-order perturbation result is -74.83eV, namely, in error by 5.28%. The problem here is that the electron-electron repulsion is an extremely large perturbation. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Variational Method Since nature will adjust the electron distribution about the nuclei so as to minimize the total energy, the electron distribution predicted by real wavefunction yields the lowest energy of the system. Trial eigenfunction 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Example: Using the Variational method to compute the ground-state energy of a helium atom 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University If we use as the trial eigenfunction, we obtain If we use as the trial eigenfunction, we obtain 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Problem: Using the trial eigenfunction for and otherwise, compute the variational energy for a particle of mass m in an infinite potential well of width . is the normalization constant. 2019/2/22 Chemistry Department of Fudan University

Chemistry Department of Fudan University Atomic Spectra Transition Matrix Elements Selection rules What is the selection rule of a electron in one dimension box? 2019/2/22 Chemistry Department of Fudan University