Day 06

the forms of the radial function R(r) and the angular wave γ(θ, Φ) for a one-electron, hydrogen-like atom

(i) the angular part of the wave function for an s orbital (1/4π)^(1/2), is always the same, regardless of the principal quantum number (ii) the angular parts of the p and d orbitals are also independent of the quantum number n therefore all orbitals of a given type (s, p, d, f) have the same angular behavior they all have the same shape (same angular part), all spherical, but differ in the sizes (different radial part)

a conceptual model for multielectron atoms

1. the quantum mechanical wave function for a multielectron atom can be approximated as a superposition of orbitals, each bearing some resemblance to those describing the quantum states of the hydrogen atom each orbital in a multielectron atom describes how a single electron behaves in the field of nucleus under the average influence of all the other electrons 2. the total energy of an atom with N electrons has the general form Eatom = F - G where F represents a sum of orbital energies F = ε1 + ε2 + ε3 + ... + εN, and G takes account of electron-electron repulsions in general, the orbital energies increase as n increases and, for equal values of n, increase as l increases 3. the order in which we assign electrons to specific orbitals is based on minimizing Eatom orbitals that minimize the value of F may not necessarily minimize Eatom ∴ be careful not to place too much emphasis on the energies of the orbitals themselves

total number of ml

2l + 1

assigning quantum numbers

by specifying three quantum numbers in a wave function Ψ, we obtain an orbital from the solutions of the Schrödinger equation for the hydrogen atom, the three quantum numbers have been obtained 1. principal quantum number, n 2. angular momentum quantum number, l 3. magnetic quantum number, ml the combination of quantum numbers produce different orbitals

quantum numbers for the first four levels of orbitals in the hydrogen atom

each orbital can hold up to to electrons

energy of orbitals in a multi-electron atom

electron-electron repulsion exists no longer degenerate due to electron-electron repulsion

multielectron atoms

in multielectron atoms, there is mutual repulsion between electrons this factor has to be considered to solve (approximate) the Schrödinger equation for this many-particle problem the results obtained are that the orbitals obtained are of the same types as those for the hydrogen atom the angular parts of the orbitals of a multielectron atom are unchanged, but the radial parts are different

spherical polar coordinate system

in spherical polar coordinate system, r is the distance of the point from the nucleus, and the angles θ and Φ describe the orientation of the distance line, r, with respect to the x, y, and z axes when this coordinate system is used, the orbitals can be expressed in terms of one function R that depends only on r, and a second function γ that depends on θ and Φ R(r) is called radial wave function and γ(θ, Φ) is the angular wave function each orbital has three quantum numbers to define it (∵ 3-D system) by specifying the 3 quantum numbers in an equation, we obtain the orbital

shielding/screening

inner electrons shield outer electrons from experiencing the full strength of nuclear charge Na 1s(2)2s(2)2p(6)3s(1) think about the attractive force of the nucleus for the electrons at different distances (from the nucleus) more distant; cannot affect the closer the closer can affect the distant electrons in orbitals closer to the nucleus screens/shields the nucleus from electrons far away decreases the actual nucleus charge felt by more distant electron Z = +11 Zeff = effective nucleus charge

angular momentum quantum number, l

l = 0, 1, 2, 3, ..., n-1 may be zero or a positive integer, but not > n-1 shape of atomic orbitals (sometimes called a subshell) the number of subshells in a principal shell is the same as the number of allowed values of the l

magnetic quantum number, ml

ml = -l, (-l + 1), ... -2, -1, 0, 1, 2, ..., (l - 1), +l may be negative or positive integer including zero orientation of the orbital in space relative to the other orbitals in the atom the number of orbitals is the same as the number of allowed values of ml for the particular value of l

orbital energy-level diagrams for the hydrogen atom and a multielectron atom (showing n = 1, 2, and 3 only)

more negative energy; closer to the nucleus multielectrons have more force, so attracts electrons more for the hydrogen atom, orbitals within a principal shell, for example, 3s, 3p, and 3d, have the same energy and are said to be energetically degenerate in a multielectron atom, orbitals within a principal shell have different energies in general, for a multielectron atom ,orbital energies increase with the value of n and for a fixed value of n, with the value of l note that the energy of a given orbital (e.g., 1s) decrease as the atomic number, Z, increases

to obtain the wave function for a particular state, we simply

multiply the radial part by the angular part

principal quantum number, n

n = 1, 2, 3, 4, ... positive, nonzero integer size and energy of the orbital the higher the value of n, the greater the electron energy and the farther, on average, the electron is from the nucleus

energy of orbitals in a single electron atom

only in single electron system degenerate; have the same energy level ground-state; highest stability, lowest energy at other orbitals; excited state

orbit/orbital

orbit is 1-D orbital is 3-D with volume

three representation of the electron probability density for the 1s orbital

recall that orbitals are wave functions, mathematical solutions of the Schrödinger wave equation Ψ^2 is a quantity related to probability density distributions (a) the probability density is represented by the height above the xy plane (b) a contour map of the 1s orbital probability density in the xy plane, pointing out the 95% contour (c) a reduced scale 3D representation of the 95% contour of a 1s orbital

d orbitals

representations of the five d orbitals 95% probability surfaces red: positive wave function blue: negative wave function first and last directed along the axes rest are between the axes

p orbitals

representations of the three 2p orbitals mutually perpendicular to one another 95% probability densities red: positive wave function blue: negative wave function

the Schrödinger equation

solutions to the Schrödinger equation for the hydrogen atom give not only energy levels but also wave functions these wave functions are called orbitals wave functions are most easily analyzed in terms of the three variables required to define a point with regard to the nucleus

principal shells and subshells

subshell ⊂ principal shell within each principal shells in the hydrogen atom, different subshells are degenerate; have the same energy value l do not matter in single electron systems orbital energies for a hydrogen atom depend only on n

Schrödinger equation can only be solved exactly for

the hydrogen atom for multi-electron systems, approximation must be made for solution (because of mutual repulsion between electrons)

extending the particle-in-a-box model to a 3-D box

the particle can move in all directions (x, y, and z) and the quantization of energy is described by the following equation there is one quantum number for each direction thus, three quantum numbers are needed in a 3-D system

orbitals of the hydrogen atom

the probability densities of the orbitals of the H atom can be represented as surfaces that encompass most of the electron probability each type of orbital has its own distinctive shape

electron spin: a fourth quantum number

the three quantum numbers (n, l, ml) provide a description of electron orbitals there are two possibilities for electron spin and this requires a 4th quantum number, the electron spin quantum number, ms how the electron is oriented never at rest; always moving: generate magnetic field

s orbitals

three-dimensional representations of the 95% electron probability density for the 1s, 2s, and 3s orbitals red: positive wave function blue: negative wave function

quantum theory of the hydrogen atom

with the ideas from wave mechanics, a conceptual model for understanding the hydrogen atom is developed this provides the basis for understanding multielectron atoms, the organization of elements in the periodic table, and, ultimately, the physical and chemical properties of the elements and their compounds in 1927, Erwin Schrödinger proposed an equation for the hydrogen atom that incorporated both the particle and the wave nature of the electron the Schrödinger equation is a wave equation that must be solved to obtain the energy levels and wave functions needed to describe a quantum mechanical system