Gaussian Definition and 772 Threads

Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70. It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16. Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.

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  1. I

    Uncertainty of mean

    I thought that maybe it would be a good idea to do gaußian error propagation of the formular of the mean, this should give me the uncertainty of the average i calculate from the sample i have... And additionally consider the standard deviation Can someone maybe give me a detailed way to handle...
  2. GLD223

    Is this a coincidence that this looks like a Gaussian?

  3. R

    B Row reduction, Gaussian Elimination on augmented matrix

    Hi! Please, could you help me on how to solve the following matrix ? I need to replace the value 3 on the third line by 0, the first column need to remain zero and 1 for the third column. I'm having a lot of difficulties with this. How would you proceed ? Thank you for your time and help...
  4. Sciencemaster

    I Can a pulse from a laser be treated as a Gaussian?

    I know a laser is generally treated as a coherent state (i.e. a plane wave). However, if a laser turns on and back off quickly, can the resulting light be treated as a gaussian multiplied by the plane wave that would have been emitted if the laser were continually active?
  5. M

    I Grating Resolving Power of Laser Beams with Gaussian Distribution

    All resources I’ve found for grating resolving power assume uniform distribution on the grating and produce airy disks. Resolvance is determined by the Rayleigh criterion where the peak of one wavelength is at the minima of the adjacent one. This definition doesn’t seem applicable for Gaussian...
  6. A

    Probability involving Gaussian random sequences

    How do I approach the following problem while only knowing the PSD of a Gaussian random sequence (i.e. I don't know the exact distribution of $V_k$)? Or am I missing something obvious? Problem statement: Thoughts: I know with the PSD given, the autocorrelation function are delta functions due...
  7. Euge

    POTW Gaussian Integrals in Two Dimensions

    Let ##a, b##, and ##c## be real numbers such that ##a## and ##c## are positive and ##ac > b^2##. Evaluate the double integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ax^2 - 2bxy - cy^2}\, dx\, dy$$
  8. J

    B Sigma Multiplied Gaussian Distribution

    Hi! Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously? Is there an equation that easily determines that? Also for other combinations like...
  9. A

    Uncertainty of the Standard Deviation

    Using this error propagation formula: I expressed the standard deviation (s) and the partial derivatives of s w.r.t. each data point as: This gives me an uncertainty of: , where m is the mean. Does this seem reasonable for the uncertainty of the standard deviation? I also found the thread...
  10. V

    Gaussian Elimination of Singular Matrix with partial pivoting

    Part (A): The matrix is a singular matrix because the determinant is 0 with my calculator. Part (B): Once I perform Gauss Elimination with my pivot being 0.6 I arrive at the last row of matrix entries which are just 0's. So would this be why Gauss Elimination for partial pivoting fails for this...
  11. E

    What is the difference between DFT and TD-DFT in Gaussian 09 software?

    Homework Statement:: difference between DFT and TDDFT in Gaussian 09 software? Relevant Equations:: hello , I hope this message finds you well Please I want to know what is the difference between DFT and time‐dependent density functional theory TD-DFT and why we use it in the measurement of...
  12. Sciencemaster

    I Calculate Gaussian Curvature from 4D Metric Tensor

    I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
  13. B

    C# How close to Gaussian a 2D Matrix percentage is in C#

    Does anyone know a C# class that can return a value (0 - 100 percentage) of How close a perfect gaussian curve an 2D Matrix is? for example, these would all return a 100%:
  14. A

    Gaussian cylinder enclosing cylinder of charge

    I am confused why we don't take into account the lids of the cylinder since the Gaussian cylinder is of finite height L as shown in the image
  15. MevsEinstein

    B Is this identity containing the Gaussian Integral of any use?

    I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?
  16. B

    I A few questions about doing a Gaussian Fit

    A few questions about doing a Gaussian Fit : 1) Is gaussian fit and gaussian regression the same thing ? 2) I have a gaussian function that will return a list of gaussian numbers giving an initial list length. So if you input 5 you will get: 1,2,6,4,1. My question is if I have an image and I...
  17. J

    Electric Field Inside a Gaussian Surface with Point Charge q

    If I have a point charge q right outside of a gaussian surface, it makes sense that the flux is zero inside the surface because the electric field going in equals the electric field going out. However, how would the electric field be zero inside? Wouldn't it just take on the electric field of...
  18. K

    I How Is Uncertainty Calculated for the Mean of a Gaussian Function in MATLAB?

    Hello! If I have N points (x,y) which I know they are described by a Gaussian i.e. y(x) is a Gaussian of unknown mean and standard deviation, and each y has an associated error of ##\sqrt{y}##, is there a general formula for the uncertainty on the mean of this function? Thank you!
  19. V

    I Diverging Gaussian curvature and (non) simply connected regions

    Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...
  20. Haorong Wu

    I Could a Gaussian beam be described as parallel at a large distance?

    Hello, there. Suppose a Gaussian beam is sent and is received at a great large distance, i.e., the propagation distance ##z \gg z_R## the Rayleigh distance. The Gaussian beam can be described by $$E_0 \frac {1}{w(z)} \exp \left ( \frac {-r^2}{w(z)^2}\right )\exp\left ( -i\left (kz+k\frac...
  21. D

    Compute the integral of the Gaussian

    why does it say transforms? is there more than one Fourier transform?? we learned in class that the inverse Fourier transform of the Fourier transform of ##f## is ##f##, so there should be just one right? I'm uncertain of how to calulate this integral though.. Mr Wolfram showed me an indefinite...
  22. M

    I Bayesian Stats: Resources about Mercer's Theorem for Gaussian Processes

    Hi, Question(s): 1. Are there any good resources that explain, at a very simple level, how Mercer's theorem is related to valid covariance functions for gaussian processes? (or would anyone be willing to explain it?) 2. What is the intuition behind this condition for valid covariance...
  23. C

    PDB file using GAUSSIAN 16 optimization

    Hello There, I am studying docking of ligand molecules into DNA using Autodock Vina. Before doing that I optimize the ligand molecule using Gaussian 16. I want to know how can we get the PDB format after optimization. Do I just need to open the .chk file and save it as PDB format or do I need...
  24. ohwilleke

    I Why Do Physicists Use Gaussian Error Distributions?

    David C. Bailey. "Not Normal: the uncertainties of scientific measurements." Royal Society Open 4(1) Science 160600 (2017). How bad are the tails? According to Bailey in an interview, "The chance of large differences does not fall off exponentially as you'd expect in a normal bell curve," and...
  25. karush

    MHB Gaussian Probability: 1 to 2 in tikz

    \begin{tikzpicture}[scale=0.6] %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All plots: 50 samples, smooth, no marks...
  26. karush

    MHB Probability of z > 1.28: Visualizing the Gaussian Curve

    ok I was able to get the graph of P(z>1.28) \begin{tikzpicture} %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All...
  27. karush

    MHB Tikz for Gaussian Function: Enhance Your Figures with Custom Commands

    ok I don't think MHB will process a newcommand but I don't know how to put this in the after \begin{tikzpicture} line the problem with posting pic here is eventually they get remove and OP is useless... this tikz code renders in overleaf but I also have many newcommands in preamble
  28. C

    I Implementation of Correlated Gaussian Random Fields Model

    Hello everyone. I have been recently working in an optimization model in the presence of uncertainty. I have read https://www.researchgate.net/publication/310742108_Efficient_Simulation_of_Stationary_Multivariate_Gaussian_Random_Fields_with_Given_Cross-Covariance in which, a methodology for...
  29. C

    MATLAB Generation of a Gaussian a random process with Matlab

    Hello everyone. I am currently working with Matlab. I have a 2D gaussian kernel constructed using the muKL technique (first attached figure). I want to use it to generate realizations of a gaussian random process using the KL theorem. For that, I obtain then all eigenvectors and eigenvalues of...
  30. D.S.Beyer

    I Discussing Interior Schwarzschild Proper Lengths & Gaussian Curvature

    I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...
  31. tworitdash

    I Can a Gaussian distribution be represented as a sum of Dirac Deltas?

    We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...
  32. Arman777

    Mutual Information from this Gaussian Distribution

    Let us suppose we are given a Gaussian Distribution in the form of $$p(x,y) \propto exp(-\frac{1}{2}x^2 - \frac{1}{2}by^2 - cxy)$$ What are the equations that I need to use to obtain Mutual Information ?
  33. Z

    What Is the Gaussian Distribution of an Ideal Gas?

    My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$ Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log...
  34. S

    Python How to properly normalize convolution of Gaussian and Lorentzian

    I'd like to plot the normalized convolution of a Gaussian with a Lorentzian (see the definitions in terms of full width half maximum (fwhm) in the attached image). Here is my attempt, but the print statements with np.trapz() do not return 1 in both cases, but rather ##\approx##0.2. I'd also like...
  35. JD_PM

    Checking convergence of Gaussian integrals

    a) First off, I computed the integral \begin{align*} Z(\lambda) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}-\frac{\lambda}{4!}x^4\right) \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}\right) \exp\left(...
  36. tanaygupta2000

    Normalize the Gaussian wave packet

    For normalizing this wave function, I began by finding the complex conjugate of psi and then multiplied it with the original psi. Now what I am getting is A^2 integral exp(2cx^2-4ax) dx = 1 Now I am not getting how to solve this exponential term. I tried by completing the square method but it is...
  37. PainterGuy

    Gaussian kernel in image processing

    I was reading the following webpage on Gaussian kernel but couldn't understand few details: https://www.imageeprocessing.com/2014/04/gaussian-filter-without-using-matlab.html . Would really appreciate it if you could guide me. Thanks in advance! Here you can find the high-res screenshot of the...
  38. A

    MHB How to Solve a Multivariate Gaussian Integral with Arbitrary Limits?

    I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $\times$ 1 and T is a positive definite symmetric p $\times$ p matrix. The integral is the...
  39. S

    I Why doesn't sinc(x) converge to Gaussian upon repeated convolution?

    Hello, I've read that repeated convolution tends, under certain conditions, to Gaussian distribution. I found this description helpful, and Wikipedia's version of this says: The central limit theorem states that if x is in L1 and L2 with mean zero and variance ##σ^2##, then...
  40. murshid_islam

    I Gaussian integral by differentiating under the integral sign

    Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...
  41. S

    Python Rotate 2D Gaussian given parameters a, b and c

    I have a Gaussian function of the form: def f(x,y): a=some number b=... c=... return 3*np.exp(-a*(-0.5 + x)**2+b*(x-0.5)*(y-0.5)-c*(-0.5 + y)**2)This is a Gaussian function symmetric around y=x, and I'd like to rotate it 45 degrees (counter)clockwise. Wikipedia gives an overdetermined...
  42. F

    Total movement of bacteria assuming a random distribution

    Hello, I have to find an expression for the total movement of a bacteria ##s##, knowing that the bacteria is placed (centered) on a two side ruler at position ##x=0## (so a negative ##x## value means the bacteria has moved to the left of the ruler) and that the probability it moves to ##x## is...
  43. M

    MHB Calculate Gaussian Quadrature: x1, x2 & w1, w2

    Hey! 😊 If we want to calculate the nodes $x_1, x_2$ and the weight functions $w_1, w_2$ for the Gaussian quadrature of the integral $$\int_{-1}^1f(x)\, dx\approx \sum_{j=1}^2w_jf(x_j)$$ is there a criteria that we have to consider at chosing the weight functions? I mean if we use e.g...
  44. patric44

    Solving a Gaussian integral using a power series?

    hi guys i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series : $$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$ to evaluate the Gaussian integral as its series some how slimier ...
  45. mertcan

    A What is the Derivation of the Inverse Gaussian Distribution by Schrödinger?

    Could you help me about the derivation of inverse gaussian distribution? During my search I encountered that it was derived by schrödinger as a result of differential equation solution but I can not find his derivation on internet...
  46. B

    A Can this difficult Gaussian integral be done analytically?

    Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...
  47. S

    Polarization and intensity of a Gaussian beam

    The equation above (from Wikipedia), assumed that the Gaussain beam has polarization in x-direction, as I know that the polarization means that the oscillation direction of the electric field and so the intenisty... so how we get circular intensity in every direction which means in x and y...
  48. C

    MHB Inequality involving Gaussian integral

    I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?
  49. JorgeM

    Gaussian wavepacket as a solution of the Schrödinger equation

    The Schrödinger equation I need to prove is this one And the Gaussian wavepacket is found here Thanks for your advice. JorgeM <Moderator's note: upload images to PhysicsForums. Do not use external image servers.>
  50. B

    Traveling quantum Gaussian wave packet

    I tried plugging Psi into the right of the Schrodinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?
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