Presentation on theme: "Lecture 1 THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM"— Presentation transcript:
1 Lecture 1 THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 1) Theoretical description of the hydrogen atom:Schrödinger equation: HY = EYHH is the Hamiltonian operator (kinetic + potential energy of an electron)E – the energy of an electron whose distribution in space is described byY - wave functionYY* is proportional to the probability of finding the electron in a given point of spaceIf Y is normalized, the following holds:The part of space where the probability to find electron is non-zero is called orbitalWhen we plot orbitals, we usually show the smallest volume of space where this probability is 90%.Comments on the Schrodinger equation:
2 2) Solution of the Schrödinger equation for the hydrogen atom 2) Solution of the Schrödinger equation for the hydrogen atom. Separation of variables r, q, fFor the case of the hydrogen atom the Schrödinger equation can be solved exactly when variables r, q, f are separated likeY = R(r) Q(q) F(f)Three integer parameters appear in the solution,n (principal quantum number), while solving R-equationl (orbital quantum number), while solving Q-equationml (magnetic quantum number), while solving F-equationn = 1, 2, … ∞; defines energy of an electronl = 0, … n-2, n-1 (total n values); defines shape of an orbital: 0-s, 1-p, 2-d, 3-f ,...ml = -l, -l+1, …, 0, …, l-1, l (total 2l+1 values); defines spatial orientationThe number of all possible combinations of l and ml for a given n isSolution of the Schrodinger equation for the case of the hydrogen atom:
3 Plot of the hydrogen atom orbitals s orbitalsp orbitalsd orbitals
4 3) The hydrogen atom orbitals Radial components of the hydrogen atom wavefunctions look as follows:R(r) = for n = 1, l = 0 1s orbitalR(r) = for n = 2, l = 0 2s orbitalR(r) = for n = 2, l = 1, ml = 0 2pz orbitalR(r) = n = 3, l = 0 3s orbitalR(r) = n = 3, l = 1, ml = 0 3pz orbitalHere a0 = , the first orbit radius (0.529Å). Z is the nuclear charge (+1)
5 4) Radial components of the hydrogen atom orbitals The exponential decay of R(r) is slower for greater nAt some distances r where R(r) is equal to 0, we have radial nodes, n-l-1 in total1s2s2pnode
6 5) Radial probability function Radial probability function is defined as [r R(r)]2 – probability of finding an electron in a spherical layer at a distance r from nucleusRadial probability plots for hydrogen atom:1s2p2s
7 Angular components of the hydrogen atom wave functions, QF for l = 0, ml = 0; QF = s orbitalsfor l = 1, ml = 0; QF = pz orbitalsfor l = 2, ml = 0; QF = dz2 orbitalsFor each orbital we have l angular nodal surfacesTherefore, together with radial nodes we have in total (n-l-1)+l = n-1 nodes for each orbitalHydrogen-like orbitals (animated):
8 7) Plot of the hydrogen atom orbitals Orbitals are called gerade (g) if they have center of symmetry or ungerade (u) if they do not have itExamples of orbitals which are gerade: s, d. Orbitals which are ungerade: p.
9 8) Allowed energies of an electron in the hydrogen atom Note:E does not depend on either l or mlIn other words, for each given ns, p, d etc orbitalsof the hydrogen atom areof the same energy (degenerate)Spectrum of hydrogen atom:4s, 4p, 4d, 4f3s, 3p, 3d2s, 2px, 2py, 2pz1s
10 SummaryExact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation;Three quantum numbers n, l, ml appear as integer parameters while solving the equation;Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (ml), number of nodal surfaces (n-1) and symmetry (g, u).