About p orbitals.
I read that those nodal planes of p and d orbitals vanish when one introduce relativistic effects. That is, the relativistic p orbital has not orbital planes. I did not the calculations for verifying this!
However, there is a typical error of interpretation of wave functions in usual textbooks. The interpretation of wavefunction phy is probabilistic, i.e. P = (phy)2 is the probability of finding the particle at a point.
The usual interpretation of p-orbitals, particles in a box (note that there are nodes inside the box for the corresponding wavefuntions) or tunnel effects is not correct one. A previous proffesor mine said that the problem of the p-lobes was a “paradox” of quantum mechanics. There is no paradox. I discuss next only p orbitals by brevity but the discussion could be adapted to the other cases.
The electron is not a part of time in a lobe and after of a time it is in the other lobe. First, that is not the interpretation of the stationary Schrödinger equation H*phy = E*phy. Second, in quantum mechanics there is not trajectories, then the electron cannot be here and after there, passing by this point or plane. There are not trajectories!
Really the P says us that the electron is “distributed” over a region or volume. Forget the physicist point of view and take a more chemical view. This is easy to see from the density charge, density = e*P, with e the electron charge in your favorite units. Then one can see half one electron in a lobe (density = e/2) and other half one in the other lobe, with a total charge of e in all the space, because the total probability is 1.
According to Schrödinger, this interpretation is not correct because one cannot measure fractions of charge, i.e. the electron is an indivisible particle. One cannot detect 0.5 electrons here and 0.5 electrons in that other region. But one would remember that p orbitals say that succeed before the measure process. When one measures electron position the wavefunction collapses to a Dirac delta function; that is, all the charge (e) is concentrated in an only point. We then detect a punctual particle with charge e.
It is somewhat as a water cloud, disperses in the sky. When some tiny part of it collapses, then makes a little sphere of water. The analogy is water cloud <=> Orbital and water sphere <=> electron as a particle.
I believe that this interpretation of electron as a diffuse cloud that collapse is more close to reality.
With regards to order filling of 4s and 3d, some textbooks are mistaken as it is claimed that 3d level has a lower energy than 4s for elements as scandium, but the real configuration is 4s2 3d1. The details of this are computational (quantum chemistry). The configuration of first transitions elements is
Sc 4s2 3d1
Ti 4s2 3d2
V 4s2 3d3
Cr 4s1 3d5
Mn 4s2 3d5
Fe 4s2 3d6
Co 4s2 3d7
Cu 4s2 3d9
Zn 4s1 3d10
The case of nickel turns out to be very interesting. According to many chemistry and physics textbooks the configuration of this element is given as 4s2 3d8. However the research literature on atomic calculations (see below) always quotes the configuration of nickel as 4s1 3d9.
Note: a Dirac function, D, can be see as the limit of an exponential
D(x) = limit [exp(-(x2)/a]
a --> 0
C.W. Bauschlicher, P. Siegbahn, and G.M. Petterson. The Atomic States of Nickel. Theoretica Chimica Acta 74: 479–491, 1988.