## Schrödinger equation for - Two-electron atoms.
- Multi-electron atoms.
## Helium-like atoms. - Singlet and triplet states.
- Exchange energy.
## What is probability of simultaneously finding a particle 1 at (*x1,y1,z1*), particle 2 at (*x2,y2,z2*), etc. => need *joint probability distribution.*
*N*-particle system is therefore a function of *3N* coordinates:
## (*x1,y1,z1; x2,y2,z2; … xN,yN,zN*) ## Must solve (1) ## First consider two particles which do not interact with one ## another, but move in potentials *V1 *and *V2. *The Hamiltonian is ## (2) ## The electron-nucleus potential for helium is ## The eigenfunctions of *H1 *and *H2 *can be written as the product:
**System of non-interacting particles** ## Using this and Eqns. 1 and 2, ## That is, where *E = E1+E2.*
## The product wavefunction* *is an eigenfunction of the complete Hamiltonian *H, *corresponding to an eigenvalue *E *which is the sum of the energy eigenvalues of the two separate particles. ## For *N-*particles, ## Eigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus ## Can be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction.
**Application to helium** ## Assuming each electron in helium is non-interacting, can assume each can be treated independently with hydrogenic energy levels: ## Total energy of two-electron system in ground ## state (*n*(1) = *n*(2) = 1) is therefore
**System of interacting particles** ## For He-like atoms can extend to *include electron-electron interaction*: ## The final term represents electron-electron repulsion at a distance *r12.* ## Or for *N-*electrons, the Hamiltonian is: -
## and the corresponding Schrödinger equation is again of the form ## where
**Wave function for system of interacting particles** ## The solutions to the equation ## can again be written in the form ## and therefore have the same analytical form as for the hydrogenic one-electron atom. ## Allowable solutions again only exist for -
## where *Zeff = Z - nl.* *Zeff *is the effective nuclear charge and *nl *is the shielding constant. This gives rise to the *shell model *for multi-electron atoms.
**Atoms with two valence electrons** ## Includes He and Group II elements (e.g., Be, Mg, Ca, etc.). Valence electrons are *indistinguishable*, i.e., not physically possible to assign unique positions simultaneously. ## This means that multi-electron wave functions must have *exchange symmetry:*
## which will be satisfied if ## That is, exchanging labels of pair of electrons has no effect on wave function. ## The “+” sign applies if the particles are *bosons. *These are said to be *symmetric *with respect to particle exchange. The “-” sign applies to *fermions,* which are *anti-symmetric *with respect to particle exchange. ## As electrons are fermions (spin 1/2), the *wavefunction of a multi-electron atom must be anti-symmetric with respect to particle exchange.*
**Helium wave functions** ## He atom consists of a nucleus with *Z = *2 and two electrons. ## Must now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin wave functions: ## As electrons are indistinguishable => must be anti-symmetric. See table for allowed symmetries of spatial and spin wave functions.
**Helium wave functions: spatial** ## State of atom is specified by configuration of two electrons. In ground state, both electrons are is 1s shell, so we have a 1s2 configuration. ## In excited state, one or both electrons will be in higher shell (e.g., 1s12s1). Configuration must therefore be written in terms of particle #1 in a state defined by four quantum numbers* *(called **). State of particle #2 called **. ## Total wave function for a excited atom can therefore be written: ## But, this does not take into account that electrons are indistinguishable. The following is therefore equally valid: ## Because both these are solutions of Schrödinger equation, linear combination also a solutions: ## where is a normalisation factor.
**Helium wave functions: spin** ## There are two electrons => S = s1+ s2 = 0 or 1. S = 0 states are called *singlets *because they only have one *ms *value*. S = *1 states are called *triplets* as *ms = *+1, 0, -1.
## There are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry. ## Only one possible *anitsymmetric spin eigenfunction:*
## There are three possible *symmetric spin eigenfunctions:*
**Helium wave functions: spin** ## Table gives spin wave functions for a two-electron system. The arrows indicate whether the spin of the individual electrons is up or down (i.e. +1/2 or -1/2). ## The + sign in the symmetry column applies if the wave function is symmetric with respect to particle exchange, while the - sign indicates that the wave function is anti-symmetric. ## The Sz value is indicated by the quantum number for *ms*, which is obtained by adding the *ms* values of the two electrons together.
**Helium wave functions** ## Singlet and triplet states therefore have different spatial wave functions. ## Surprising as spin and spatial wavefunctions are basically independent of each other. ## This has a strong effect on the energies of the allowed states.
**Singles and triplet states** ## Physical interpretation of singlet and triplet states can be obtained by evaluating the *total spin angular momentum *(S), where * *is the sum of the spin angular momenta of the two electrons.
## The magnutude of the total spin and its *z*-component are quantised: ## where *ms = -s, … +s *and *s = *0, 1.* * ## If *s1 = +*1/2 and *s2 = *-1/2 => *s = *0. - Therefore
*ms = *0 (*singlet state)*
## If *s1 = +*1/2 and *s2 = *+1/2 => *s = *1. - Therefore
*ms = -*1, 0, +1 (*triplet states*)
* *
**Helium terms** ## Angular momenta of electrons are described by *l1, l2, s1, s2.*
## As *Z<*30 for He, use *LS *or *Russel Saunders coupling.*
## Consider ground state configuration of He: 1s2 ## Orbital angular momentum: *l1=l2 = *0 => L = *l1 + l2 = *0 ## Spin angular momentum: *s1 = s2 = *1/2 => S = 0 or 1 - Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (
*singlet*) or 2(1) + 1 = 3 (*triplet*)
## J = L + S, …, |L-S| => J = 1, 0. - Therefore there are two states: 11S0 and 13S1 (also using
*n = *1)
## But are they both allowed quantum mechanically?
**Helium terms** ## Must consider *Pauli Exclusion principle*: “*In a multi-electron atom, there can never be more that one electron in the same quantum state”*;* *or equivalently, “*No two electrons can have the same set of quantum numbers”.*
## Consider the 11S0 state: L = 0, S = 0, J = 0 *n1 = *1*, l1 = *0, *ml1 =* 0, *s1 = *1/2, *ms1 = *+1/2 *n2 = *1*, l2 = *0, *ml2 =* 0, *s2 = *1/2, *ms2 = *-1/2
## 11S0 is therefore *allowed* by Pauli principle as *ms *quantum numbers differ. ## Now consider the 13S1 state: L = 0, S = 1, J = 1 *n1 = *1*, l1 = *0, *ml1 =* 0, *s1 = *1/2, *ms1 = *+1/2 *n2 = *1*, l2 = *0, *ml2 =* 0, *s2 = *1/2, *ms2 = *+1/2
## 11S1 is therefore *disallowed* by Pauli principle as *ms *quantum numbers are the same.
**Helium terms** ## First excited state of He: 1s12p1 ## Orbital angular momentum: *l1= *0, *l2 = *1 => L* = *1 ## Spin angular momentum: *s1 = s2 = *1/2 => S = 0 or 1 - Multiplicity (2S+1) is therefore 2(0) + 1 = 1 or 2(1) + 1 = 3
## For L = 1, S = 1 => J = L + S, …, |L-S| => J = 2, 1, 0 ## Therefore have, *n1 = *1, *l1 = *0, *s1 = *1/2 ## and *n2 = *2, *l2 = *1, *s1 = *1/2 ## For L = 1, S = 0 => J = 1 - Term is therefore 1P1
- Allowed from consideration of Pauli principle
**Helium terms** ## Now consider excitation of both electrons from ground state to first excited state: gives a 2p2 configuration. ## Orbital angular momentum: *l1 = l2=*1 => L = 2, 1, 0 - Produces S, P and D terms
## Spin angular momentum: *s1 = s2= *1/2 = > S =1, 0 and multiplicity is 3 or 1 ## *Violate Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P)
**Helium Grötrian diagram** ## Singlet states result when S = 0. ## Triplet states result when S = 1
**Exchange energy** ## Need to explain why triplet states are lower in energy that singlet states. Consider ## The expectation value of the Hamiltonian is ## The energy can be split into three parts, *E = E1 + E2 + E12*
* *where
## The expectation value of the first two terms of the Hamiltonian is just ## where *ER = *13.6 eV is called the *Rydberg energy.*
**Exchange energy** ## The third term is the electron-electron Coulomb repulsion energy: ## where the + sign is for singlets and the - sign for triplets and *D *is the direct Coulomb energy and *J *is the *exchange Coulomb energy:*
## The resulting energy is *E12 ~ *2.5 *ER. *Note that in the exchange integral, we integrate the expectation value of *1/r12 *with each electron in a different shell. See McMurry, Chapter 13.
**Exchange energy** ## The total energy is therefore ## where the + sign applies to singlet states (S = 0) and the -sign to triplets (S = 1). ## Energies of the singlet and triplet states differ by 2*J. *Splitting of spin states is direct consequence of exchange symmetry. ## We now have, *E1 + E2 = -*8*ER *and *E12 = *2.5*ER => E = -*5.5*ER* = -74.8 eV ## Note: - Exchange splitting is part of gross structure of He - not a small effect. The value of 2
*J *is ~0.8 eV. - Exchange energy is sometimes written in the form
- which shows explicitly that the change of energy is related to the relative alignment of the electron spins. If aligned = > energy goes up.
**Helium terms** ## Orthohelium states are lower in energy than the parahelium states. Explanation for this is: - Parallel spins make the spin part of the wavefunction symmetric.
- Total wavefunction for electrons must be antisymmetric since electrons are fermions.
- This forces space part of wavefunction to be antisymmetric.
- Antisymmetric space wavefunction implies a larger average distance between electrons than a symmetric function. Results as square of antisymmetric function must go to zero at the origin => probability for small separations of the two electrons is smaller than for a symmetric space wavefunction.
- If electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy.
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