[QE-users] What is the difference between ILDOS and |psi|^2?

Thu Dec 19 03:10:57 CET 2024

Dear Professors Stefano Baroni and Stefano de Gironcoli,

Thank you for your quick responses.

I don't know the basics of spherical harmonic functions, so I'm going to study them.

By the way, it seems that the explanation so far is based on the premise that the p-orbital is degenerate, but what do you think?

When degeneracy is released, the energies of each orbital (px, py, pz) are different, and the contribution of each can be different. 

In such a case, the sum of the probability densities can be written as:

∣ψ∣total2∝cxx2×fx(r)+cyy2×fy(r)+czz2×fz(r)|\psi|^2_{\text{total}} \propto c_x x^2 \times f_x(r) + c_y y^2 \times f_y(r) + c_z z^2 \times f_z(r)∣ψ∣total2∝cxx2×fx(r)+cyy2×fy(r)+czz2×fz(r)

Here, cx, cy, and cz represent the contribution weights of each orbital, and fx(r), fy(r), and fz(r) may be different from each other even if they are spherically symmetric. In such a case, the sum of the probability density is generally considered to not have spherical symmetry.

This is because x, y, and z have different symmetries.

How do you think about it.

Best regards,

Toshiharu Higuchi

From: Stefano de Gironcoli <degironc at sissa.it> 
Sent: Wednesday, December 18, 2024 10:54 PM
To: toshi1higu at docomo.ne.jp; 'Quantum ESPRESSOusers Forum' <users at lists.quantum-espresso.org>; 'Stefano Baroni' <baroni at sissa.it>
Subject: Re: [QE-users] What is the difference between ILDOS and |psi|^2?

you may want to read and understand the basics of spherical harmonics, for instance from the wikipedia page

https://en.wikipedia.org/wiki/Spherical_harmonics

in particular see the attached bit.

Stefano deGironcoli

On 18/12/24 09:47, toshi1higu at docomo.ne.jp <mailto:toshi1higu at docomo.ne.jp>  wrote:

Dear Professor Stefano Baroni,

Thank you for your quick response.

I think LDOS & ILDOS are defined as,

LDOS=Σ|psi|^2  * δ(E-En,k)

and  ILDOS=∫(LDOS) dE

I don't understand the relationship between the above ILDOS and what you said, "x^2 + y^2 + z^2 is spherically symmetric."

Sorry, but could you please explain?

Best regards,

Toshiharu Higuchi

From: Stefano de Gironcoli  <mailto:degironc at sissa.it> <degironc at sissa.it> 
Sent: Wednesday, December 18, 2024 5:09 PM
To:  <mailto:toshi1higu at docomo.ne.jp> "toshi1higu at docomo.ne.jp"  <mailto:toshi1higu at docomo.ne.jp> <toshi1higu at docomo.ne.jp>; Quantum ESPRESSOusers Forum  <mailto:users at lists.quantum-espresso.org> <users at lists.quantum-espresso.org>; Stefano Baroni  <mailto:baroni at sissa.it> <baroni at sissa.it>
Subject: Re: [QE-users] What is the difference between ILDOS and |psi|^2?

  x^2 + y^2+ z^2 is spherically symmetric, isn't it ?

Stefano

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  _____  

From: users <users-bounces at lists.quantum-espresso.org <mailto:users-bounces at lists.quantum-espresso.org> > on behalf of Toshiharu Higuchi via users <users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org> >
Sent: Wednesday, December 18, 2024 8:38:54 AM
To: users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org>  <users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org> >; Stefano Baroni <baroni at sissa.it <mailto:baroni at sissa.it> >
Subject: Re: [QE-users] What is the difference between ILDOS and |psi|^2? 

Dear Professor Stefano Baroni,
Thank you for your quick response.
The |psi|^2 (square of absolute value of wave function) of the 2p orbital is 2-lobes, and the |psi|^2 of the 3d orbital is 4-lobes, so they are not spherically symmetric. I don't quite understand the relationship between this and what you said, "px, py, and pz are degenerate, and their square coefficients are spherically symmetric." 
Sorry, but could you please explain?
Best regards,
Toshiharu Higuchi

-----Original Message-----
From: Stefano Baroni <baroni at sissa.it <mailto:baroni at sissa.it> > 
Sent: Wednesday, December 18, 2024 3:38 PM
To: "toshi1higu at docomo.ne.jp <mailto:toshi1higu at docomo.ne.jp> " <toshi1higu at docomo.ne.jp <mailto:toshi1higu at docomo.ne.jp> >; "users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org> " <users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org> >
Subject: Re: [QE-users] What is the difference between ILDOS and |psi|^2?

Dear Toshiharu: The ILDOS sums the squared moduli of all the molecular orbitals in a given energy range. px, py, and pz are degenerate, and their square modulus spherically symmetric (can you see why?) … Can you now draw the conclusions yourself? Stefano

___
Stefano Baroni, Trieste -- http://stefano.baroni.me

> On 18 Dec 2024, at 03:56, Toshiharu Higuchi via users <users at lists.quantum-espresso.org <mailto:users at lists.quantum-espresso.org> > wrote:
> 
> Dear QE Users
> I have a question about the difference between ILDOS and |psi|^2.
> When I calculated the ILDOS (plot_num=10) and |psi|^2 (plot_num =7) of 
> a hydrogen atom, the patterns were different.
> In the case of |psi|^2, 1s orbital → spherical, 2p orbital → 2-lobes, 
> and 3d orbital → 4-lobes were obtained.
> However, in the case of ILDOS, all patterns were spherical.
> According to the literature, ILDOS can be calculated from |psi|^2 and 
> Dirac's delta function.
> Even if |psi|^2 is a pattern other than spherical, ILDOS will be spherical.
> I don't understand the reason for this.
> Please tell me,
> (1) In Quantum-ESPRESSO, are ILDOS and |psi|^2 calculated in separate 
> processes? Or is |psi|^2 calculated first and then ILDOS is calculated 
> using this value?
> (2) Does Dirac's delta function have any effect?
> 
> Best regards,
> Toshiharu Higuchi
> 
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