- #1
hokhani
- 483
- 8
- TL;DR Summary
- The non spherical orbitals of spherical Hamiltonian
Oritals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?
hokhani said:Summary:: The non spherical orbitals of spherical Hamiltonian
Oritals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?
hokhani said:How is this possible?
That's true of any central force problem, even classically. For example the Sun's gravitational field is (too good approximation) spherically symmetrical but the solar system obviously is not.hokhani said:Orbitals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?
The reason is that the named orbitals commonly drawn are basis-dependent. It is like drawings of a circle from different (basis-dependent) perspectives, which produces unsymmetric ellipses.hokhani said:Orbitals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?
What is the source of atoms?hokhani said:Suppose that an electron is in a d-orbital, say ##d_{z^2}## . The probability of existence of electron in one direction may be different from that in another direction! I think this discrepancy can be explained as follows:
There is no preferred z-direction and the shape of orbitals helps to determine the dynamics of electrons.
The shape of atomic orbitals is determined by the mathematical equations that describe the probability of finding an electron at a particular location around the nucleus of an atom. These shapes can be visualized as three-dimensional regions where the electron is most likely to be found.
Atomic orbitals are named based on their shape and orientation. The letters s, p, d, and f correspond to the different shapes of orbitals: spherical, dumbbell, cloverleaf, and complex, respectively. The numbers 1, 2, 3, etc. indicate the energy level of the orbital, with 1 being the lowest energy level.
The number of atomic orbitals in each energy level is equal to the square of the principal quantum number (n). For example, the first energy level (n=1) has 1 orbital, the second energy level (n=2) has 4 orbitals, and the third energy level (n=3) has 9 orbitals.
An s orbital is spherical in shape and can hold a maximum of 2 electrons. A p orbital is dumbbell-shaped and can hold a maximum of 6 electrons. Additionally, p orbitals have three different orientations (px, py, and pz) while s orbitals only have one.
The shape of atomic orbitals plays a crucial role in chemical bonding. For example, the overlap of two s orbitals results in a sigma bond, while the overlap of two p orbitals results in a pi bond. The orientation of these orbitals also determines the type of bond formed. Additionally, the number and arrangement of electrons in orbitals determine the overall shape of a molecule, which affects its chemical properties.